Dynamics of 2D Potts and random-cluster models

by

Reza Gheissari

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Mathematics

New York University

May 2019

Eyal Lubetzky

Charles Newman

Acknowledgements

Firstly, I am indebted to Eyal Lubetzky for his tremendous generosity with his

time, experience, and knowledge. I am grateful he first introduced me to the field

of Markov chain mixing times and its links with equilibrium statistical mechanics.

More importantly, he has always been someone I could look up to when needing

motivation, talk to when seeking advice (academically or otherwise), bounce ideas

off of when feeling stuck, and run to when excited about any incremental progress.

I truly could not have had a better mentor, advocate, and role-model, and I will

always count myself fortunate that we joined Courant at the same time.

I would like to thank Chuck Newman for serving as a constant source of

inspiration and mentorship throughout my five years at Courant. He has taught me

much of what I know of two-dimensional statistical physics and percolation theory,

and his door has always been open when I have needed someone to look to for help.

I am grateful for my interactions and collaboration with Gerard Ben Arous. He

is a source of unbounded energy and ideas, and I am lucky to have learned about

mean-field spin glasses from him. I am also thankful for his wisdom and availability

the many times I have turned to him for guidance and direction.

This would not have been possible without the good fortune of having Clement

Hongler as an REU advisor at Columbia. He introduced me to mathematical re-

search through the beautiful field of two-dimensional statistical physics at criticality,

put up patiently with my mathematical immaturity, and strongly encouraged me

to pursue a PhD in mathematics at Courant. I also thank Julien Dubedat for his

time and tremendous patience while serving as my undergraduate thesis advisor.

Indeed being at Courant has been one of the best journeys of my life, and the

Courant community has been a most warm, welcoming, and inspirational group. I

iii

owe everything I know to the faculty members from whom I have learned so much

at Courant and at Columbia. I thank Dan Stein for his availability through our

many conversations and collaborations. And I have been lucky to attend insightful

courses from all of the aforementioned, as well as Professors Ofer Zeitouni, Yuri

Bakhtin, Paul Bourgade, Afonso Bandeira, Lai Sang Young, Georg Stadler, Joel

Spencer, Dorian Goldfeld, Daniela De Silva among others.

I especially thank Yuri Bakhtin, Paul Bourgade, Eyal Lubetzky, Chuck Newman,

and Professor Varadhan for agreeing to be on my thesis committee.

Besides the faculty, my time at Courant has also been shaped by my peers,

including Aukosh, Moumanti, Lisa, Krishnan.

I am grateful to my close friends: Danny, Ty, Emily, Nathalie, Sindhu, Erin,

Sarah, Trey, Sabrina, and many others. They have supported me, kept me grounded,

and made sure I take the time to step away from mathematics and enjoy life.

Most importantly, I thank my family for their unconditional love and support.

To my sister, Roya, I have been lucky to have you nearby; you have always been

there, both to talk through difficult decisions with, and as a source of distraction

from mathematics. To my parents, I cannot describe the immense role you have

played in making me the person I am. Thank you for always pushing me, giving

me honest advice, and believing in me; and thank you for instilling in me the

importance of a deep and unending yearning for knowledge.

iv

Abstract

This thesis considers the dynamical phase portrait of the Potts and random-

cluster (FK) models in two-dimensions, as the system size diverges.

The Potts model was introduced in the 1950’s as a model of ferromagnetism

where interacting particles take q ≥ 2 possible states (q = 2 being the Ising model).

The random-cluster (FK) model is a dependent percolation model indexed by a

real-valued q ∈ (0,∞) (q = 1 being independent percolation); it is central to the

analysis of the Potts models, as at integer q it encodes the correlations of the

corresponding Potts model. In 2D, the models have a sharp transition between

a high-temperature phase and a low-temperature phase, via rich behavior at a

critical point: at criticality, when q ≤ 4, they are conjectured to have a conformally

invariant scaling limit with random fractal interfaces (SLE’s), while when q > 4

they are characterized by phase coexistence and Brownian interfaces.

We are interested in understanding how long canonical Markov chains like the

Glauber dynamics for the Potts and FK models, as well as cluster dynamics such

as Swendsen–Wang, take to approximate the equilibrium measure on, say, an n× n

box. These dynamics are well-known to equilibrate rapidly at all high temperatures,

but are believed to undergo critical slowdowns, depending on the order of the phase

transition: this was only known for q = 2 where they mix in polynomial time (in n),

and for sufficiently large q, where they take exponentially long to equilibrate.

We begin by showing that the following holds for the mixing times (as well as

inverse spectral gaps) of Potts Glauber dynamics on the n×n torus, (Z/nZ)2. The

Potts Glauber dynamics at criticality has a mixing time that is polynomial in n

when q = 3, and nO(logn) when q = 4; on the other hand, for every q > 4 the mixing

time of the critical Potts dynamics grows exponentially in n. We also fill in the

v

rest of this phase diagram by proving that for every q, as soon as the temperature

is sub-critical, the Potts Glauber dynamics becomes exponentially slow.

The FK Glauber dynamics on (Z/nZ)2 (as well as the related Swendsen–Wang

dynamics) are known to be fast mixing O(log n) at all off-critical temperatures. We

prove that at the critical point, their mixing time on (Z/nZ)2 is at most nO(logn)

for all q ∈ (1, 4] whereas it becomes exponentially slow in n as soon as q > 4.

The analysis of the FK Glauber dynamics and Swendsen–Wang dynamics is

substantially complicated by their non-locality; long-range interactions can be

encoded into the FK boundary conditions on an n× n box. We first show that in

all off-critical regimes, the mixing time is stable—up to polynomial factors—with

respect to choice of realizable FK boundary conditions. We then turn to the critical

point, where we show that when q is large, the dynamics are sensitive to the choice

of boundary condition. For free or monochromatic boundary conditions and large q,

these dynamics at criticality are much faster than on the torus and mix in exp(no(1))

time. We conclude by investigating this sensitivity at criticality, under boundary

conditions that interpolate between free, wired, and periodic.

vi

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction 1

1.1 Glauber dynamics for the Potts model . . . . . . . . . . . . . . . . 8

1.2 Glauber dynamics for the FK model . . . . . . . . . . . . . . . . . 12

1.3 Dependence on FK boundary conditions off-criticality . . . . . . . . 18

1.4 Sensitivity to boundary conditions in coexistence regimes . . . . . . 23

1.5 Some open problems . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Preliminaries 32

2.1 The Potts and FK models . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Markov chain mixing times . . . . . . . . . . . . . . . . . . . . . . . 42

3 Equilibrium estimates at q ∈ [1, 4] and RSW Theory 59

3.1 Crossing probabilities and RSW Theory . . . . . . . . . . . . . . . 60

3.2 Boundary bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Mixing on (Z/nZ)2 at continuous phase transitions 95

vii

4.1 Potts Glauber dynamics at continuous phase transition points . . . 96

4.2 FK Glauber dynamics at continuous phase transition points . . . . 103

5 Slow mixing in phase coexistence regions 134

5.1 Potts Glauber dynamics on (Z/nZ)2 in phase coexistence regions . 136

5.2 FK dynamics on (Z/nZ)2 at a discontinuous phase transition . . . . 141

5.3 Potts Glauber dynamics with free boundary at β > βc(q) . . . . . . 145

6 FK boundary conditions in uniqueness regions 148

6.1 Fast mixing on thin rectangles . . . . . . . . . . . . . . . . . . . . . 155

6.2 Polynomial mixing time for realizable boundary conditions . . . . . 184

6.3 Near-optimal mixing time for typical boundary conditions . . . . . 193

6.4 A canonical paths bound with realizable FK boundary conditions . 202

6.5 Slow mixing under worst-case boundary conditions . . . . . . . . . 211

7 Boundary conditions in coexistence regions I: free and monochro-

matic boundary 224

7.1 Cluster expansion and equilibrium estimates . . . . . . . . . . . . . 227

7.2 Upper bounds under free boundary conditions . . . . . . . . . . . . 236

8 Boundary conditions in coexistence regions II: other boundary

conditions 262

8.1 Subexponential mixing time with Dobrushin boundary conditions . 274

8.2 Sub-exponential mixing on cylinders . . . . . . . . . . . . . . . . . 281

8.3 Slow mixing with phase-symmetric boundary conditions . . . . . . . 292

Bibliography 297

viii

List of Figures

1.1 The Potts phase transition on Z2 . . . . . . . . . . . . . . . . . . . 4

1.2 Critical Potts configurations at q = 3, 4, 5 . . . . . . . . . . . . . . . 6

1.3 Dynamical phase diagram for the Potts Glauber dynamics . . . . . 11

1.4 A critical FK configuration with nested long-range connections . . . 13

1.5 Long-range FK boundary connections . . . . . . . . . . . . . . . . . 14

1.6 Dynamical phase diagram for the FK Glauber dynamics . . . . . . 15

1.7 Swendsen–Wang with free and periodic boundary . . . . . . . . . . 25

1.8 Dependence of Swendsen–Wang dynamics on boundary conditions

in phase coexistence regimes . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Bounding q = 4 crossing probabilities . . . . . . . . . . . . . . . . . 64

3.2 A pair of boundary connections, called bridges, separated by a

dual-connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 A pair of boundary bridges, γi, γi+1, over e ∈ ∂nR induced by a

configuration on Λ−R, and separated by a dual-bridge over e. . . . 74

3.4 Construction for the upper bound on |Be1| . . . . . . . . . . . . . . 83

3.5 Construction for the upper bound on |Be2| . . . . . . . . . . . . . . 88

ix

4.1 Coupling beyond a dual-crossing, with the appropriate bridges dis-

connected in the FK model . . . . . . . . . . . . . . . . . . . . . . . 116

4.2 Block choices for the block dynamics when proving mixing time

upper bounds on the torus . . . . . . . . . . . . . . . . . . . . . . . 127

5.1 Geometric bottleneck for phase coexistence on the torus: Potts . . . 137

5.2 Geometric bottleneck for phase coexistence on the torus: FK . . . . 143

6.1 Long-range FK boundary connections necessitate recursion on groups

of rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.2 Some examples of disconnecting intervals . . . . . . . . . . . . . . . 160

6.3 The subsets used to recurse on groups of rectangles . . . . . . . . . 171

6.4 The presence of certain dual-connections allows us to couple FK

configurations inside disconnect intervals . . . . . . . . . . . . . . . 175

6.5 Choice of subsets to bound the FK mixing time under realizable

boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.6 A spatial mixing estimate near “typical” FK boundary conditions . 196

7.1 The choice of the censored FK Glauber dynamics allows us to couple

configurations started from all-wired and all-free . . . . . . . . . . . 241

7.2 Enlargements En(Q) and E′n(Q) of a rectangular subset Q . . . . . 245

7.3 Boundary modification to restrict the influence of the open cluster

of the southern boundary . . . . . . . . . . . . . . . . . . . . . . . . 247

7.4 Boundary modification to couple the two FK configurations on the

central column C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

x

8.1 Some of the sets and interfaces considered in the study of tilted FK

interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

8.2 Bounding the probability that a tilted FK interface undergoes a

deviation of at least ±h . . . . . . . . . . . . . . . . . . . . . . . . 266

8.3 The diagonal strips used to construct a censored FK dynamics . . . 277

8.4 The process by which the censored dynamics pushes the FK interface

down to its equilibrium location . . . . . . . . . . . . . . . . . . . . 279

8.5 The constituent events that push the FK interface from one side to

the other on a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 288

xi

Chapter 1

Introduction

Background

The Ising model was introduced in the 1920’s by Lenz and Ising as a model of

the physical phenomenon of ferromagnetism, where a slab of iron spontaneously

magnetizes after coming in brief contact with an external magnetic field. The

mathematical model consists of a random assignment of ±1-spins to the vertices of a

graph describing the underlying atomic structure of the ferromagnet; the distribution

is governed by a competition between the system’s internal energy, which favors

local alignment, and its entropy, moderated by an inverse temperature β > 0.

In 1951, Domb and Potts generalized the Ising model to the q-state Potts model,

in which particles may take on q ≥ 2 possible states (with q = 2 corresponding to

the Ising model). The Ising and Potts models have seen tremendous attention (cf. [4,

71, 95]), being exactly solvable in the 1D and mean-field settings, and exhibiting

rich phase transitions on the physically relevant square lattices Zd. As such, since

their introductions, the models have become prototypes for understanding sharp

phase transitions and critical phenomena in statistical physics. They have also

1

found applications to many other areas, ranging from ecology and computational

biology, to statistical inference, image processing, and the theory of neural networks.

Let us now define the Potts model more precisely. The Potts model on a graph

G = (V,E) at inverse-temperature β > 0 is the distribution µG,β,q over colorings

of the vertices of G with q colors (states) amongst [q] = 1, ..., q, in which the

probability of a configuration σ ∈ [q]V is given by

µG,β,q(σ) ∝ exp[−βH(σ)] , where H(σ) :=∑

(x,y)∈E

1σx 6= σy , (1.0.1)

i.e., the Hamiltonian H(σ) counts the number of pairs of adjacent vertices that have

distinct colors (see §2.1.1). As mentioned, the Ising/Potts models are of particular

interest in the cases where their underlying geometry has a physically relevant

lattice structure, e.g., on the square lattice Zd (d ≥ 2). Here, they are believed

to undergo a rich phase transition at some critical temperature βc(q), separating

a high-temperature disordered phase, where correlations decay rapidly, from a

low-temperature phase where a macroscopic long-range order emerges.

Fortuin and Kasteleyn introduced the random-cluster (FK) model in 1972 [36];

the FK model on a graph G = (V,E) with parameters 0 < p < 1 and q > 0 is the

distribution πG,p,q over edge subsets of G, where the probability of an edge-subset ω

with o(ω) edges and k(ω) connected components (in the subgraph (V, ω)) satisfies

πG,p,q(ω) ∝ [p/(1− p)]o(ω)qk(ω) , (1.0.2)

(see also §2.1.2). The model generalizes Bernoulli bond percolation (q = 1) and has

far-reaching connections to electrical networks and random spanning trees (q ↓ 0):

see [47]. The FK model is also closely related to the Ising/Potts models at integer

2

q ≥ 2, with connectedness in ω encoding the correlations of the corresponding

Potts model via the Edwards–Sokal coupling [35]; e.g., one may produce σ ∼ µG,β,q

by first sampling ω ∼ πG,p,q for p = 1 − e−β, then assigning (all vertices of)

each connected component in ω a uniformly chosen color amongst [q] = 1, .., q

independently. As such, extensively studied in its own right, the random-cluster

representation has been an important tool in the analysis of Ising and Potts models

(see e.g., [5, 32, 34] for a sampling of breakthrough results on the Potts model that

rely on its random-cluster representation).

The canonical Markov chains for the Potts and FK models are the local-update

Glauber dynamics (e.g., heat-bath, Metropolis) and the non-local Swendsen–Wang

dynamics; these are important both as Markov Chain Monte Carlo samplers from

the Potts and FK measures, and because of the insights they give regarding

the physical off-equilibrium evolution of the corresponding statistical mechanics

systems [45, 94]. Over the years, there has been much effort in physics, theoretical

computer science, and mathematics communities to connect the fascinating static

phase transition of such spin systems to their off-equilibrium dynamics, measured

by e.g., autocorrelation length, spectral gap, or mixing time.

This thesis is part of this mathematical program, analyzing the off-equilibrium

dynamics of the Potts and FK models asymptotically as the system size diverges.

Specifically, we focus on understanding how the equilibrium phase transitions

of Potts and FK models present in their dynamical phase portrait as various

parameters (q, the temperature p or β, and the boundary conditions) are varied.

We study this in the particularly rich and physically relevant setting where the

underlying geometry is planar, e.g., a subset of the integer lattice Z2.

The results in this thesis are primarily a compilation of joint works with

3

Figure 1.1: Potts configurations for q = 3 at β < βc (left), β = βc (middle), andβ > βc (right) on the torus (Z/nZ)2.

Lubetzky [41, 42, 43] and a joint work with Blanca and Vigoda [8].

Potts and FK models on Z2

When the underlying graph is Z2, or some finite subset of Z2, and q ≥ 1,

significant progress has been made in the study of these models’ equilibrium

behavior, characterized by a sharp phase transition at a critical pc(q) (respectively,

βc(q), again related via the relation p = 1− e−β).

Using a duality relation between random-cluster models on planar graphs G

and on their planar dual G∗, and the self-duality of the integer lattice Z2, it

has long been predicted that the critical point of the FK model on Z2 is its self-

dual point psd(q) =√q

1+√q, (and therefore the critical point of the Potts model

is βsd(q) = log(1 +√q)). Indeed this has been known to be the critical point

of the Ising model (q = 2) dating back to the work of Kramers and Wannier in

1941 [56]. In the setting of two-dimensional Bernoulli bond percolation (q = 1),

Kesten (1980) proved [54] that the critical point is given by pc(1) = 12

which

matches the self-dual point psd(1). Later, using very different tools, namely the

4

celebrated techniques of reflection positivity and cluster expansion, the critical point

was identified with the self-dual point for sufficiently large q [55] (this region where

cluster expansion converges has since been brought down to q > 24.78 [57, 58]).

More recently, the seminal work of Beffara and Duminil-Copin [5] identified the

equality pc(q) = psd(q) =√q

1+√q

(and therefore βc(q) = log(1 +√q)) for all q ≥ 1.

The phase transition occurring at pc(q) and βc(q) in Z2 can be characterized

in various different ways. One canonical such choice is described in terms of the

infinite-volume measures on Z2, which can be obtained by taking weak limits of

the measures on Gn ↑ Z2 (see §2.1 for the consistency relations required on an

infinite-volume measure). In the case of the FK model, at high temperatures

p < pc(q), under πZ2,p,q, almost surely every connected component of ω is finite,

whereas at low temperatures p > pc(q), under πZ2,p,q there almost surely exists an

infinite connected component in ω. In the case of the Potts model, this translates

to the following: at high temperatures β < βc(q), there is exactly one unique

infinite-volume Gibbs measure on Z2, whereas at low-temperatures when β > βc(q)

there are multiple distinct infinite-volume Gibbs measures (consisting of mixtures of

q extremal measures corresponding to the situations in which each of the q possible

colors is dominant). In particular, when p < pc(q) (resp., β < βc(q)), probabilites

of FK connections (resp., correlations between spins) decay exponentially fast in

their Euclidean distance, whereas when p > pc(q) (β > βc(q)) they stay uniformly

bounded away from zero. (See Fig. 1.1 for a depiction in the case q = 3.)

Perhaps the most important question in the equilibrium analysis of these two-

dimensional statistical physics models is then to identify the behavior at the critical

pc(q) or βc(q). In particular, the phase transitions can be classified as either

first-order (discontinuous) or second-order (continuous). If the phase transition

5

Figure 1.2: Critical Potts configurations at q = 3, 4, 5 on the torus (Z/nZ)2 atn = 64 with FK boundaries between different clusters.

is second-order the FK representation almost surely does not have any infinite

clusters at its critical point, and the spin-system has a unique infinite-volume

Gibbs measure at β = βc; if the phase transition is first-order, however, the Potts

model has multiple Gibbs measures and the FK model almost surely has an infinite

cluster at criticality (at least under some choice of an infinite-volume FK Gibbs

measure). The famous solution of Onsager (1944) [75] showed that the Ising model

exhibits a continuous phase transition at βc(2); the result of Kesten [54] showed

the same for Bernoulli percolation on Z2; on the other hand, when q > 24.78, the

phase transition is first-order where a “disordered” high-temperature-like phase

coexists with a low-temperature-like “ordered” phase so that the FK model has

two extremal infinite-volume measures, giving rise to q+ 1 extremal infinite-volume

Potts measures.

Recently, the breakthrough of [34] combined sophisticated tools from the theory

of discrete holomorphic observables with percolation and Russo–Seymour–Welsh

(RSW) techniques to prove the continuity of the phase transition at q ∈ [1, 4].

This was followed by [32] which established that as soon as q > 4, the FK and

Potts models have a first-order phase transition as described above. At continuous

phase transition points, the behavior is believed to be governed by conformal field

6

theory (CFT) [6, 27], exhibiting remarkable features like power-law correlation

decay, conformal invariance, and convergence of interfaces to random fractals known

as Schramm–Loewner evolution (SLE) (this has been established in the special

case of q = 2, as well as q = 1 on the triangular or hexagonal lattices). On the

other hand, at discontinuous phase transition points, it is characterized by phase

coexistence and surface tension that is reminiscent of its low-temperature behavior,

but with the addition of the metastable “disordered” phase. See also Fig. 1.2.

In what follows, we build on this recent work to fill in a corresponding dynamical

phase diagram for the planar Potts and FK models, considering canonical Markov

chains like the Glauber and Swendsen–Wang dynamics. The central objects of

study for the dynamics are the spectral gap and mixing time (i.e., measuring

the time it takes for the Markov chain to converge to its invariant distribution).

In two-dimensions, these dynamics are of added interest at the critical point

itself, where the mixing time depends heavily on the order of the phase transition,

and is predicted to exhibit dynamical critical phenomena, e.g., dynamic critical

exponents, universality, and non-trivial scaling limits (see, e.g., the well-known

physics review [49]).

In §1.1 we consider the Glauber dynamics for the Potts model on an n×n torus

(Z/nZ)2 at its critical point βc(q) in the three regimes: 1 < q < 4, the extremal

q = 4, and q > 4. In §1.2, we consider the Glauber dynamics for the FK model on

(Z/nZ)2 in those same regimes, as well as the intimately related Swendsen–Wang

dynamics of the Potts model. The analysis of the FK dynamics is considerably more

intricate due to the presence of long-range dependencies in the Glauber dynamics’

transition rates. These long-range dependencies can be encoded into the boundary

conditions, and we study how they can affect the mixing times first in the setting of

7

p 6= pc(q) for the FK dynamics in §1.3. In §1.4, we turn to analyzing the sensitivity

of mixing times for FK and Swendsen–Wang dynamics to boundary conditions, in

the phase coexistence regime of p = pc(q) and q large: we classify the mixing times

under boundary conditions including (and interpolating between) free, wired and

toroidal. We end the introduction in §1.5 with a selection of related open problems

in the understanding of Potts and FK dynamics on Z2.

1.1 Glauber dynamics for the Potts model

Glauber dynamics is a local Markov chain, introduced by Glauber (1963) in [45],

modeling the physical evolution of a spin system following a deep quench, i.e.,

initialized from some off-equilibrium configuration. Various flavors of Glauber

dynamics (heat-bath, Metropolis) are also some of the most frequently used Markov

chain Monte Carlo (MCMC) samplers for such spin systems. For the Potts model,

the heat-bath Glauber dynamics updates each vertex via an i.i.d. rate-1 Poisson

process, where its new value is sampled according to µG,β,q conditioned on the

values of all other vertices: that is, it is the continuous-time Markov process that,

when the clock at a vertex i rings updates the spin σi according to the following

transition rates:

µG,β,q(σi = ` | σG\i

)=

eβ∑

(i,j)∈E 1σj=`∑1≤`≤q e

β∑

(i,j)∈E 1σj=`, for ` = 1, ..., q .

Its discrete-time version at each integer time-step picks a vertex uniformly at

random, then updates its state according to the same transition rate. The Potts

Glauber dynamics on G is, by construction, ergodic and reversible with respect to

the Potts distribution µG,β,q, and therefore one is interested in the time it takes for

8

the dynamics to approximate its invariant measure.

The spectral gap of a discrete-time Markov chain, denoted gap, is 1−λ where λ

is the largest nontrivial eigenvalue of the transition kernel, and for a continuous-time

chain it is the gap in the spectrum of its generator. The inverse of the spectral

gap, denoted gap−1 serves as an important gauge for the rate of convergence of the

chain to equilibrium in L2. The mixing time, tmix, of a Markov chain is a closely

related quantity (see (2.2.3)) which is defined by the time it takes to be within

L1-distance of at most ε of stationarity, from a worst-case initialization.

The static phase transition of the Ising, Potts and FK models are believed to

manifest in their Glauber dynamics on n× n boxes or torii. At this point a rich

dynamical phase transition has been established in the case of the Ising model: at

high temperature (β < βc(2)) the inverse-gap gap−1 is O(1) and the system rapidly

mixes in L1 at time Cβ log n+O(1) [1, 62, 65, 68, 69], whereas at low temperature

(β > βc(2)), these slow down to exp(cβn+ o(n)) [19, 20]. At the critical point, the

mixing time is believed to be nz for some dynamic critical exponent, which has

seen extensive numerical and exact study in physics communities and is believed to

be universal [25, 59, 74, 93]. Lubetzky and Sly [64] showed that the mixing time

indeed has a polynomial critical slowdown, though the rigorous upper and lower

bounds are far apart.

In the case of the Potts model, the absence of monotonicity and a more limited

understanding of the equilibrium measure made the corresponding picture less

understood. With the identification of the critical point [5], the exponential decay

of correlations at high temperatures implied an order one gap and optimal O(log n)

mixing [2, 3, 69] for all β < βc(q). Besides the natural expectation that the mixing

is slow for all β > βc(q), physicists predicted that the mixing time at criticality is

9

polynomial (with some dynamic critical exponent) when q = 3, 4 in accordance with

the continuous phase transition [25, 26, 31], whereas it should slow to exponential

in n as soon as q > 4. However, this was only known rigorously for the perturbative

regimes of (1) sufficiently large β βc(q) and (2) all β ≥ βc(q) for sufficiently large

q 1 [12, 13].

The first results of this thesis are to fill in much of the remainder of this

dynamical phase diagram on (Z/nZ)2 (see Figure 1.3 for a visualization).

Theorem 1. There exist c1, c′1 > 0 and c2, c

′2 > 0 so that continuous-time Glauber

dynamics on (Z/nZ)2 satisfy the following: for the 3-state critical Potts model,

nc1 . gap−1p . nc

′1 , (1.1.1)

and for the 4-state critical Potts model,

nc2 . gap−1p . nc

′2 logn . (1.1.2)

In fact, Eqs. (1.1.1)–(1.1.2) hold for arbitrary boundary conditions on an n× n box.

Theorem 2. Let β and q be such that the Potts model has two distinct infinite-

volume measures on Z2, i.e., µ1Z2,β,q 6= µ2

Z2,β,q (corresponding to two different

dominant colors). There exists c(β, q), c′(β, q) > 0 such that Glauber dynamics for

the q-state Potts model at β on (Z/nZ)2 satisfy

c ≤ 1

nlog gap−1

p ≤ c′ . (1.1.3)

In particular, this implies slow mixing on (Z/nZ)2 in the following two regimes:

10

β

q

β

q

Figure 1.3: The dynamical phase portrait of the Potts Glauber dynamics on (Z/nZ)2

as β, q vary (the grey curve indicating β = βc(q)). On the left, a visualizationof what was known, and on the right, our contribution. Red indicates the high-temperature regime with an order one inverse spectral gap, orange indicates apolynomial one, and shades of blue indicate an inverse gap that is exponential in n.

1. Low temperature: For every q, if β > βc(q), then 1n

log gap−1p 1.

2. Discontinuous phase transition: If q > 4 and β = βc(q), then 1n

log gap−1p 1.

Crucial to the analysis of [64] of the critical Ising dynamics (q = 2) were Russo–

Seymour–Welsh (RSW) estimates for the corresponding FK model—stating that

on n×m rectangles with uniformly bounded aspect ratios and arbitrary boundary

conditions, crossing probabilities are uniformly bounded away from zero—obtained

by [33] using the discrete holomorphic observable framework of Smirnov [88]. The

framework of [88] is further applicable to the critical Potts model for q = 3, and

the above RSW-type estimates for the FK-Ising model were extended [34] to this

case; this allows us to similarly extend the dynamical analysis of [64] to q = 3 and

obtain (1.1.1). However, at q = 4, these RSW estimates only hold in the bulk of

the n×m rectangle, and crossing probabilities near the boundary are believed to

be sensitive to boundary conditions; this difficulty results in our quasi-polynomial

(rather than a polynomial) upper bound on mixing (1.1.2).

When β > βc(q), or β = βc(q) and q > 4, the phase coexistence implies that

11

crossing probabilities and related quantities depend on which phase is “picked out”

to be dominant; in Section 5.1, we introduce a novel geometric bottleneck based on

multiple crossings of non-trivial homology on the torus to derive the implication

that phase coexistence implies slow mixing of Glauber dynamics on (Z/nZ)2. We

are also able to show slow mixing for the Potts Glauber dynamics as soon as

β > βc(q) on an n× n box with no (free) boundary conditions in Section 5.3.

1.2 Glauber dynamics for the FK model

We now turn to the heat-bath Glauber dynamics for the critical 2D FK model

in the same regimes of interest of q ∈ (1, 4] and q > 4. The FK Glauber dynamics

is defined in analogy to the Potts Glauber dynamics, but for edge updates. Each

edge receives an i.i.d. rate-1 Poisson clock, and when its clock rings, the inclusion

of e in the subgraph ω (also identified with an assignment ω ∈ 0, 1E) is chosen

according to πG,p,q conditioned on ω \ e: if the edge e = (x, y), then

πG,p,q(e ∈ ω | ωE\e) =

p

q(1−p)+p if x←→6 y in (V, ω \ e);

p if x←→ y in (V, ω \ e),

where x ←→ y in a graph G = (V,E) indicates x, y are in the same connected

component of G. Notice, importantly, that unlike the Glauber dynamics of the

Potts model, the transition rates for an edge e are not simply a function of the

states of its neighboring edges (or even those in an O(1) neighborhood).

All the same, the FK dynamics is believed to undergo a similar critical slowdown

phenomenon to the Potts Glauber dynamics: on an n × n torus, for all p 6= pc

(where the FK model has uniqueness) the mixing time of the dynamics has an order

12

Figure 1.4: A critical FK configuration that induces three nested, distinct boundarycomponents (red, blue, purple) called bridges.

one spectral gap and order log n mixing time, while at p = pc it should behave as

nz for some universal z > 0 in the presence of a continuous phase transition, and

as exp(cn) in the presence of a discontinuous phase transition.

For p 6= pc, the facts that tmix log n and gap−1rc 1 on (Z/nZ)2 were

established in [9] using the exponential decay of cluster diameters in the high-

temperature regime p < pc: on finite boxes with select nice boundary conditions,

this translates to a property known as strong spatial mixing, implying that the

number of disagreements between the states of two chains started at different initial

states decreases exponentially fast, thus tmix log n; this result readily extends to

p > pc by the duality of the two-dimensional FK model.

At q ∈ (1, 4] and p = pc(q), where correlations decay polynomially and not

exponentially fast, the upper bounds on gap−1p for the Ising and q = 3, 4-state Potts

models from [64] and Theorem 1 used a multi-scale approach that reduced the side

length of the box by a constant factor in each step via a coupling argument. On

(Z/nZ)2, these carry over to the FK dynamics for q = 2, 3, 4 by the comparison

estimates of [91], which we will discuss later. However, for non-integer q, FK

configurations may form macroscopic connections along the boundary of smaller-

scale boxes, destroying the coupling, because of the long-range dependencies in the

FK dynamics’ transition rates (see Fig. 1.5)—this is prevented for integer q thanks

13

Figure 1.5: Long-range connections in the boundary conditions stand in the way ofcoupling FK configurations beyond a boundary interface.

to the special relation between FK and Potts models.

To control this effect, in Section 3.2, we rely on percolation techniques and

RSW theory to prove upper and lower bounds on the total number of nested

disjoint macroscopic connections along the boundaries of smaller-scale boxes at

p = pc—these estimates may be of independent interest. We then need to devise a

new dynamical scheme relying on a coupling between a censored Markov chain with

a systematic block dynamics chain to push through the coupling while modifying

the boundary conditions at each step. Refer to §4.2.1 for a detailed proof overview.

It was recently shown [48] that for q = 2 the FK Glauber dynamics on any

graph G = (V,E) has gap−1rc ≤ |E|O(1); the technique there, however, is highly

specific to the case of q = 2. Indeed, this bound is violated on Zd, for any d ≥ 2,

at p = pc and q large, as follows from the exponential lower bounds of [12, 13], as

well as in the mean-field setting at any q > 2 [10, 38, 44, 46]. The best prior upper

bound on non-integer 1 < q < 4 was tmix ≤ exp(O(n)); we prove that for periodic

boundary conditions (as well as a wide class of others, including wired and free; see

Remark 1.2.1), the following holds:

Theorem 3. Let q ∈ (1, 4] and consider the Glauber dynamics for the critical FK

14

p

q

p

q

Figure 1.6: The phase portrait of the FK Glauber dynamics on (Z/nZ)2 as p, qvary (the grey curve indicating p = pc(q)). On the left, a visualization of what wasknown, and on the right, our contribution. Red indicates the uniqueness regimewith an order one inverse gap, brown indicates an nO(logn) one, which is expectedto be polynomial, and blue indicates an inverse gap that is exponential in n.

model on (Z/nZ)2. There exists c = c(q) > 0 such that gap−1rc . nc logn.

In the case of a discontinuous phase transition q > 4, these long-range depen-

dencies are avoided, as we no longer rely on a multi-scale argument to bound the

mixing time, using instead the FK analogue of the geometric bottleneck used to

prove Theorem 2.

Theorem 4. Let q > 4 and consider the Glauber dynamics for the critical FK

model on (Z/nZ)2. There exists c(q), c′(q) > 0 such that

c ≤ 1

nlog gap−1

rc ≤ c′ .

Refer to Figure 1.6 for a visualization of the contributions Theorems 3–4.

Remark 1.2.1. Theorem 3 holds for rectangles with uniformly bounded aspect

ratio, under any set of FK boundary conditions (formally defined in §2.1.2) with

the following property: for every edge e on the boundary of the box, there are

O(log n) distinct boundary components connecting vertices on either side of e (see

15

Definitions 4.2.11–4.2.12 and Theorem 4.2.14). This includes, in particular, the

wired and free boundary conditions, as well as, with high probability, “typical”

boundary conditions: those that are sampled from πZ2,pc,q (see Lemma 4.2.17).

1.2.1 Consequences for the Swendsen–Wang dynamics

Swendsen–Wang dynamics is a (non-local) Markov chain on Potts configura-

tions, introduced in 1987 by Swendsen and Wang [90], that exploits the coupling

between the FK and Potts models, to overcome low-temperature bottlenecks in the

energy landscape (thus providing a potentially faster sampler compared to Glauber

dynamics) via global cluster flips. Because of this, it is widely used as the standard

MCMC sampler for the Ising/Potts models.

More precisely, the Swendsen–Wang dynamics is the discrete-time Markov chain

that from a Potts configuration σ(t) generates configuration σ(t+ 1) by

1. generating an intermediate FK configuration ω ∈ 0, 1E as follows: for

e = (x, y), set ωe = 0 if σx(t) 6= σy(t), and ωe = 1 with probability p and

ωe = 0 with probability 1− p if σx(t) = σy(t).

2. to each connected component C ⊂ V of (V, e : ωe = 1), assign an i.i.d. color

ΞC ∼ Unif1, ..., q, and assign σx(t+ 1) = ΞC for every x ∈ C.

Chayes–Machta dynamics [21] is a closely related Markov chain on FK configurations,

analogous to Swendsen–Wang for integer q, yet defined for any real q ≥ 1 (see §2.2.1).

The second (re-coloring) step of the Swendsen–Wang dynamics allows it to

quickly jump between configurations in which one color is dominant, to those in

which another color is dominant; it therefore seamlessly swaps between the different

metastable phases at low-temperatures. However, as with FK Glauber dynamics,

16

the Swendsen–Wang dynamics may still be slowed down by the bottleneck between

the “ordered” and “disordered” phases, which can coexist at p = pc(q).

Ullrich formalized the close connection between the Swendsen–Wang dynamics

and the FK Glauber dynamics via the following comparison inequality.

Theorem 1.2.2 ([91, 92]). Let q ≥ 2 be integer. Let gapp and gaprc be the spectral

gaps of discrete-time Glauber dynamics for the Potts and FK models on a graph

on m edges and maximum degree ∆, resp., and let gapsw be the spectral gap of

Swendsen–Wang dynamics. Then

gapp ≤ 2q2(qe2β)4∆gapsw , (1.2.1)

gaprc ≤ gapsw ≤ 16gaprcm logm. (1.2.2)

Theorem 1.2.2 implies, in particular, that the mixing time of the FK Glauber

dynamics and Swendsen–Wang dynamics on a graph G are comparable up to

polynomial factors in the number of vertices. However, the Potts Glauber dynamics

can be much slower than both, e.g., on the n× n torus when β > βc(q).

Via these comparison estimates, [91, 92] concluded polynomial upper bounds on

the mixing times of the Swendsen–Wang dynamics at all off-critical temperatures

β 6= βc(q), as well as for the Ising Swendsen–Wang dynamics at β = βc(2). Later, [7]

proved that in fact the Swendsen–Wang dynamics has an order one spectral gap and

mixes in optimal O(log n) time when β < βc(q). It was at first unclear whether the

Swendsen–Wang dynamics undergoes a critical slowdown analogous to that of the

Ising/Potts models; Gore and Jerrum [46] showed that indeed, at the critical point

on the complete graph, for every q > 2 (where the phase transition is discontinuous)

the Swendsen–Wang dynamics has tmix ≥ exp(Ω(√n)); the full dynamical picture

17

for the dynamics on the complete graph has since been established in [10, 38, 44].

On the torus (Z/nZ)d with d ≥ 2, [12, 13] proved that for q sufficiently large, the

Swendsen–Wang dynamics becomes exponentially slow at βc(q).

These comparison inequalities allow us to translate our mixing time estimates

for the FK dynamics on (Z/nZ)2 into sharper bounds on the Swendsen–Wang

dynamics on (Z/nZ)2.

Corollary 1.2.3. Consider the Swendsen–Wang dynamics for the q-state Potts

model at its critical point β = βc(q) on (Z/nZ)2. At q = 3, it has gap−1sw = nO(1),

and at q = 4 it has gap−1sw = nO(logn). For every q > 4, it has 1

nlog gap−1

sw 1.

Remark 1.2.4. For q ∈ 2, 3, 4, the bounds of Theorem 1 for the Potts Glauber

dynamics held uniformly over all possible Potts boundary conditions. The com-

parison estimates of [91] carry the upper bounds on the mixing time of the Potts

Glauber dynamics to the FK Glauber dynamics and Swendsen–Wang dynamics,

yet only for a limited class of boundary conditions (e.g., there can be only one

boundary component of size larger than nε, in contrast to “typical” ones). More

generally, the exponential dependence in the maximum degree in (1.2.1) causes

boundary conditions with large components to be very problematic when applying

these comparison inequalities. We explore this further in Section 1.4.

1.3 Dependence on FK boundary conditions off-

criticality

While the FK representation is intimately related to the Ising/Potts models,

unlike the latter spin systems, the weight of a configuration ω is not a function of

18

local interactions between edges, but instead of global connectivity properties. This

non-local structure is a crucial feature of the model but significantly complicates its

analysis. As hinted at in the previous section, FK boundary conditions can enforce

such long-range interactions that destroy spatial mixing properties and can make

the dynamical analysis of the model much more difficult, both at criticality and

off-criticality. In this section, we carefully analyze the effect of boundary conditions

on the dynamics, in the uniqueness regime of p 6= pc(q) where the FK dynamics

exhibits optimal O(log n) mixing in the absence of boundary conditions [10].

For an n×n box Λn ⊂ Z2 with nearest-neighbor edges, we let ∂Λn be its (inner)

boundary (i.e., those vertices in Λn that are adjacent to vertices in Z2 \Λn). Recall

that an Ising/Potts boundary condition τ is a fixed assignment of spins to ∂Λn,

and µτΛn is the Gibbs distribution on Λn conditional on the assignment τ to ∂Λn.

(Since the interactions are nearest-neighbor, this is the same as conditioning on

a configuration on all of Z2 \ (Λn \ ∂Λn).) For the FK model on Λn, a boundary

condition ξ on ∂Λn is a partition ξ1, ξ2, ... of the vertices in ∂Λn such that all

vertices in ξi are forced to be in the same connected component of a configuration ω

via “ghost” (or external) wirings; these connections are considered in the counting

of k(ω) in (1.0.2) and can impose highly non-local interactions. When running the

FK Glauber dynamics in the presence of boundary conditions, the connectivity of

vertices x, y is computed with the connectivities imposed by the boundary taken

into consideration.

Of particular interest are boundary conditions for the FK model corresponding

to configurations on Z2 \ Λn: i.e., where the boundary partition is induced by

the connections of a random-cluster configuration on E(Z2) \ E(Λn). We call

such boundary conditions realizable. (In fact, many works, including the standard

19

text [47], often restrict attention to realizable boundary conditions.)

For the Ising/Potts model Glauber dynamics, the Θ(log n) mixing at high

temperatures β < βc(q) is well understood to hold independently of the boundary

conditions [2, 3, 5, 18, 69]. These bounds follow as a consequence of the exponential

decay of correlations of the model in the high-temperature regime, which holds

even near the boundary uniformly over choice of Ising/Potts boundary conditions;

this property is known as strong spatial mixing (SSM).

The corresponding bound for the FK Glauber dynamics, of Θ(log n) mixing

on Λn when p 6= pc(q), relied on a strong spatial mixing property for the random-

cluster model; however unlike the Ising/Potts models, strong spatial mixing for

the FK model only holds in the presence of boundary conditions that do not carry

information about random-cluster connectivities in non-local ways [3]: namely,

configurations in different regions of Λn do not interact through these boundaries.

As such, the Θ(log n) upper bound of [9] only holds under boundary conditions

that are free (no boundary condition), wired (all boundary vertices are connected

to one another) or periodic (the torus). The behavior of the FK Glauber dynamics

under other random-cluster boundary conditions remained unclear.

We prove that the FK dynamics at p 6= pc(q) mixes in polynomial time, uniformly

over all realizable boundary conditions.

Theorem 5. For every q > 1, p 6= pc(q), there exists a constant c(p, q) > 0 such

that the FK Glauber dynamics on the n×n box Λn ⊂ Z2 with any realizable boundary

condition ξ has gap−1rc = O(nc).

We pause to comment on the proof of Theorem 5. As mentioned above, it is

easy to construct examples of realizable boundary conditions where the correlation

between edges near the boundary does not decay in their distance, even if p pc(q),

20

as the boundary can enforce long-range interactions. Since the exponential decay of

correlations does hold for edges in the “bulk” of Λn (i.e, at distance Θ(log n) away

from its boundary), we are able to reduce the proof of Theorem 5 to proving a

polynomial upper bound for the mixing time of the FK-dynamics on thin rectangles

of dimension n×Θ(log n) with realizable boundary conditions. This will be the

key technical difficulty for us and is established in Theorem 6.1.1.

In the setting of spin systems, or boundary conditions that do not encode

long-range interactions, a polynomial upper bound on n×Θ(log n) rectangles would

follow from standard canonical paths arguments [53, 66, 67, 86]. However, even

realizable boundary conditions can heavily distort the graph with external wirings,

preventing this approach from succeeding (c.f., Section 6.4). Instead, to prove

Theorem 6.1.1 we devise a novel application of the recursive (block dynamics)

scheme that was used in the proofs of Theorems 1 and 3. We point the reader to

Section 6.1 for a more detailed proof overview.

Using Theorem 5, we prove near-optimal O((log n)C) mixing time for “typical”

boundaries as we detail now. The notion of typicality should be understood as

with high probability under some probability distribution over realizable boundary

conditions, with a natural choice being the marginal distribution of the infinite

random-cluster measure πZ2,p,q on E(Z2) \ E(Λn).

Theorem 6. For every q > 1, p < pc(q) (resp., p > pc(q)), there exists a constant

C > 0 such that if a boundary condition ξ on Λn is randomly drawn according

to πZ2,p′,q′(·E(Z2)\E(Λn)) for any p′ < pc(q′) (resp., p′ > pc(q

′)), with probability

1− o(1), the continuous-time FK Glauber dynamics on Λn with boundary condition

ξ has gap−1rc = O((log n)C).

The proof of Theorem 6 uses Theorem 5 in a crucial way. Typical boundary

21

conditions do not exhibit the strong spatial mixing property from [3, 9]; however,

for boundary conditions having boundary components of size at most α log n,

correlations between edges near the boundary decay exponentially in their graph

distance divided by α log n. Using this, we reduce bounding the mixing time on Λn

with typical boundaries to bounding the mixing time on Θ((log n)2)×Θ((log n)2)

rectangles with arbitrary realizable boundary conditions. Theorem 5 implies that

the mixing time in these smaller rectangles is at most poly-logarithmic in n.

Given that our rapid mixing result for realizable boundaries relies heavily on the

planarity of the boundary connections in Z2 \ Λn, one may wonder whether rapid

mixing holds for all possible FK boundary conditions (including those not realizable

as configurations on Z2 \ Λn). We answer this in the negative, showing that there

exist (non-realizable) boundaries for which the FK-dynamics is slow mixing even

while p 6= pc(q). In fact, this torpid mixing holds at p pc(q), which may sound

especially surprising as correlations in πΛn,p,q die off faster as p decreases.

Theorem 7. Let q > 2. For every α ∈ (0, 12] and λ > 0 there exists a boundary

condition ξ, such that when p = λn−α the FK Glauber dynamics on the n× n box

Λn with boundary condition ξ has gap−1rc = exp(Ω(nα)).

Our proof of this theorem is constructive: we take any graph G on m edges for

which slow mixing of the FK-dynamics is known at some value of p(m) < pc(q), and

show how to embed G into the boundary of Λn. We then develop a procedure to

transfer mixing time bounds from G (speficially the case where G is the complete

graph [10, 38, 44, 46]) to Λn at p(m) sufficiently small so that the effect of the

configuration away from the boundary is negligible.

We remark about the condition q > 2 in Theorem 7. In [48] it was shown

that the mixing time of FK-dynamics when q = 2 is at most polynomial in the

22

number of vertices on any graph and at every p ∈ (0, 1). It is believed that this

rapid mixing holds for all q ≤ 2; hence the requirement q > 2 should be sharp for

Theorem 7. We believe that the above slow mixing result may also extend to small,

but Ω(1) values of p < pc(q), though our current proof does not allow for this. In

principle, one would want to embed a bounded degree graph into ∂Λn, so that its

critical point at which it exhibits slow mixing is Ω(1). There are several examples

of bounded degree graphs where torpid mixing is known [12, 13, 24, 41, 42].

1.4 Sensitivity to boundary conditions in coexis-

tence regimes

In the previous section, we investigated the sensitivity of mixing times in the

uniqueness regime of the FK dynamics p 6= pc(q). Theorems 5–6 hinted at stability

of mixing times with respect to changes of boundary conditions, as long as they

are compatible with the planarity of the model.

In this section, we stick to combinations of free, wired, and periodic boundary

conditions and demonstrate how different boundary conditions can dramatically

alter the mixing time at criticality in phase coexistence regimes, namely at q > 4

and p = pc(q). None of these boundary conditions carry long-range information.

Instead, the dependence of mixing times on boundary conditions here is because

the boundary conditions can pick out one of the two dominant phases (free and

wired) and destroy the metastability of the other. Since we are now discussing FK

boundary conditions which correspond to some Potts boundary condition at integer

q (free to free, and wired to monochromatic, say, red) the results will hold for both

the FK dynamics and the Swendsen–Wang dynamics of the Potts model.

23

1.4.1 Free and monochromatic boundary conditions

A similar sensitivity to boundary conditions in phase coexistence regimes has

been extensively studied in the low temperature Ising model. There, it is a long-

standing open problem to show that the mixing time under all-plus boundary

conditions is order n2, with the largest minus component following a droplet

shrinkage picture governed by mean-curvature flow [11, 52]. Martinelli [66] showed

that under all-plus boundary conditions, the mixing time is eO(√n) at β βc as

compared with exponentially slow on the torus; this was subsequently improved to

exp(no(1)) and then nO(logn) and all β > βc in [63, 70].

If we naively followed the intuition from the low temperature Ising and Potts

model, free boundary conditions—where the q different colors’ phases are metastable

so gap−1p & exp(cn)—might be expected to induce the same (slow) critical mixing

behavior as in the torus. However, this is not case, as the following theorem

demonstrates; see also Fig. 1.7.

Theorem 8. Fix any q large enough and let p = pc(q). The following holds for the

FK Glauber dynamics on an n× n box with free boundary conditions:

gap−1rc . exp

(no(1)

). (1.4.1)

At integer q and β = βc(q), the same bound holds for the Swendsen–Wang dynamics.

The estimate (1.4.1) holds also for wired FK boundary conditions (which at

integer q correspond to monochromatic Potts boundary conditions), since the free

boundary conditions at pc(q) are self-dual to wired boundary conditions (see §2.1.2).

In fact, we establish (1.4.1) for FK boundary conditions sampled from the free or

wired infinite-volume Gibbs measures (see Proposition 7.0.2), as well as ones that

24

500 1000 1500 2000 2500 3000

0.2

0.4

0.6

0.8

1.0

periodic

free

Figure 1.7: Swendsen–Wang realizations for the critical 5-state Potts model, froma monochromatic initial state, under free (left) vs. periodic (right) boundaryconditions; plot shows largest component size in the corresponding FK configuration.

are free on three sides and wired on the fourth (Corollary 7.2.7).

Theorems 2 and 8 show the similarities between the dynamical behavior of

the FK model at its critical point pc(q) in the presence of a discontinuous phase

transition, and the 2D Ising model in the low temperature regime β > βc(2). In fact,

the proof of Theorem 8 follows the approach used in [70] to establish sub-exponential

upper bounds on tmix for Ising Glauber dynamics in the presence of all-plus boundary

conditions. The absence of monotonicity in the Potts model frequently leads us

to work directly with the FK representation. However, this entails adapting the

cluster-expansion techniques and the Wulff construction framework of [30] to the

25

FK model. Moreover, unlike the Ising model—where central to the upper bounds on

mixing in many related works is the coupling of configurations beyond an interface

between clusters (e.g., the interface between the plus and minus phases, used to

establish the inductive step in the multi-scale argument of [66])—recall that the

boundary conditions of the FK model may feature long-range connections between

vertices. Using these as a “bridge” over the interface (see Figure 1.5), different

FK configurations below the interface may induce different distributions above it,

thus preventing the coupling. Working around obstacles of this type comprises a

significant part of the proof of Theorem 8, as it did the proofs of Theorems 3 and 5.

1.4.2 Combinations of free, monochromatic, and periodic

boundary conditions

We next investigate the relationship between mixing times and boundary con-

ditions that interpolate between periodic and free/monochromatic, at the critical

point of a discontinuous phase transition (see Figure 1.8). As with Theorem 8, our

results hold for q large enough, and are expected to hold whenever the order-disorder

surface tension (see Def. 7.1.1) is positive, namely at p = pc for all q > 4 (see [32]).

Recall that ∂Λn is the set of vertices of Λn adjacent to Z2 \ Λn. Also, let ∂nΛn

be its northern boundary, and define the eastern, western, and southern boundaries

∂eΛn, ∂wΛn, ∂sΛn similarly.

Theorem 9 (Mixed b.c.). Let q be large, ε > 0, and let (an, bn, cn, dn) be marked

vertices on ∂Λn such that they are not all within εn of any one side of ∂Λn and are

all distance greater than εn from each other. There exists c(ε, q) > 0 (independent

of n and an, bn, cn, dn) such that the FK Glauber dynamics on Λn at p = pc(q) with

26

Boundary Swendsen–Wang

Periodic/Mixed

||

||

tmix ≥ exp(cn)

Dobrushin tmix ≤ exp(O(√n log n))

Cylindrical

||

||

||

||

||

||

tmix ≤ exp(n1/2+o(1))

Figure 1.8: Mixing time bounds for Swendsen–Wang dynamics on n × n boxeswith different sets of boundary conditions. Dashed lines indicate free boundaryconditions, the bold red lines denote red boundary conditions, and hash markings|, || indicate periodic boundary conditions on their respective sides. The differentboundary conditions depicted here are special cases of Theorems 4 and 8–11.

boundary conditions that are wired on the boundary segments (an, bn) and (cn, dn)

and free elsewhere, has

gap−1rc & exp(cn) .

In particular, this holds with wired boundary on ∂e,wΛn and free elsewhere.

When q is integer and β = βc(q), the same holds for the Swendsen–Wang

dynamics, where wired boundary conditions are replaced with monochromatic, i.e.,

all sites on the boundary segments (an, bn) and (cn, dn) take the same state/color.

Theorem 10 (Dobrushin b.c.). Let q be large and let (an, bn) be marked vertices

on ∂Λn. There exists c(q) > 0 (independent of n and an, bn) so that FK Glauber

dynamics at p = pc(q) with boundary conditions that are wired on the boundary

27

segment (an, bn) and free elsewhere, has

gap−1rc . exp(c

√n log n) .

In particular, this holds with wired boundary on ∂s,wΛn and free elsewhere. When q

is integer and β = βc(q), the same bound holds for Swendsen–Wang dynamics.

Theorem 11 (Cylinders). Let q be large and p = pc(q). The FK Glauber dynamics

with periodic boundary conditions on ∂n,sΛn and either wired or free boundary

conditions on each of ∂eΛn and ∂wΛn satisfies

gap−1rc . exp(n1/2+o(1)) .

When q is integer and β = βc(q), the same holds for the Swendsen–Wang dynamics.

Remark 1.4.1. Theorems 9–11 showed the dependence of mixing times on bound-

ary conditions for the critical FK model in the phase coexistence regime. If, instead,

one were interested in the simpler setting of Glauber dynamics for the 2D Ising

model at large β (where the plus and minus phases would assume the role of wired

and free phases in our theorems), the proofs would carry over (and even simplify

due to the absence of long-range interactions), via the tools of [30].

Remark 1.4.2. By the well-known relation between gap and the total variation

mixing time tmix (see (2.2.3) in §2.2), the bounds in Theorems 1–11 all hold

with gap−1 replaced by tmix. Similarly, standard comparison estimates (see [60,

Lemma 13.22]) extend our bounds for heat-bath Glauber dynamics to Metropolis

(and other flavors of local-update dynamics), and the bounds on FK Glauber

dynamics to the non-local Chayes–Machta dynamics (and its variants).

28

1.5 Some open problems

There are many interesting open questions that remain unresolved in the under-

standing of the dynamics of Ising, Potts, and FK models on the two-dimensional

lattice Z2. In the context of the Ising model, many of the biggest open problems are

very well-known (e.g., determining the precise polynomial for the mixing time at

β = βc(q), and identifying the mixing time of the low-temperature model with plus

boundary conditions as Θ(n2)). In what follows we collect some less well-established

open problems that arise specifically in the context of the results of this thesis,

regarding the dynamics of the planar Potts and FK dynamics.

Critical behavior at continuous phase transition point

Beginning with the Potts Glauber dynamics on the torus (Z/nZ)2, the most

clear open problem remaining from our results is the following:

Problem 1.5.1. Show that there exists c > 0 such that the Potts Glauber dynamics

at q = 4 and β = βc(q) on (Z/nZ)2 has gap−1p . nc.

Addressing this question requires a sharper coupling technique than the coupling

beyond a given dual-FK crossing as done in Chapter 4; this is because at q = 4

under the wired boundary conditions, the probability of such a top-to-bottom

crossing is not expected to satisfy an RSW estimate. Rather one would want

to rely on the fact that typical boundary conditions support such dual-crossings

with some positive probability, and sub-blocks of Λn will have typical boundary

conditions with high probability; the ideas in proving Theorem 3 may prove useful

for this. Indeed, pushing this approach through seems closely related to resolving

the following gap left open by Theorem 3.

29

Problem 1.5.2. Show that there exists a c(q) > 0 such that the FK Glauber

dynamics at q ∈ (1, 4] and p = pc(q) on (Z/nZ)2 has gap−1rc . nc.

In the other direction, we draw attention to the fact that there is no polynomial

lower bound on the mixing time or inverse gap of the continuous-time FK Glauber

dynamics (or Swendsen–Wang dynamics) at criticality for any q ∈ (1, 4] (including

even the case of FK-Ising model q = 2). The rigorous lower bound of Li and

Sokal [61] bounds an auto-correlation time by the static critical exponents of the

model—namely the specific heat—but in the only case where these exponents are

known rigorously, q = 2, the specific heat actually vanishes.

Question 1.5.3. Does the Swendsen–Wang dynamics for the critical Ising model

on (Z/nZ)2 exhibit polynomial slowdown? More generally, show there exist c(q) > 0

such that the continuous-time FK Glauber dynamics at q ∈ (1, 4] and p = pc(q)

have gap−1rc & nc (and similarly for gap−1

sw at q = 2, 3, 4).

Behavior in phase coexistence regimes

The latter half of this thesis turns to the behavior of the Glauber and Swendsen–

Wang dynamics in the phase coexistence regime of q > 4 and p = pc(q). In this

entire regime, in Chapter 5 we identify the exponentially slow mixing for all these

dynamics on (Z/nZ)2; we then turn to the interplay between metastability of phases,

and the ability of boundary conditions to break this metastability—however, those

results are all perturbative and only apply for q sufficiently large.

Problem 1.5.4. Show that the FK Glauber dynamics and Swendsen–Wang dynam-

ics at q > 4 and p = pc(q) on Λn with free or monochromatic boundary conditions

have a sub-exponential mixing time.

30

The above problem is primarily a question of an equilibrium nature, as it requires

various sharp estimates on order-disorder interfaces (see Section 7.1), which rely

on the perturbative cluster expansion approach, to be extended to all q > 4. An

open problem of a more dynamical nature is to translate the sub-exponential upper

bounds under free and monochromatic boundary conditions to the Potts Glauber

dynamics: at present all the dynamical tools used in Chapters 7–8 rely crucially on

the monotonicity of the FK dynamics.

Problem 1.5.5. Show that the Potts Glauber dynamics on Λn with free or

monochromatic boundary condition has gap−1p ≤ exp(no(1)) when q > 4 and

β = βc(q) (or even q sufficiently large and β = βc(q)).

In fact, as far as we know, it is even open to show for q 6= 2 a sub-exponential

upper bound on the mixing time of the Potts Glauber dynamics at low-temperatures

with monochromatic boundary conditions. One would expect a droplet shrinkage

governed by motion by mean-curvature (with a continuous-time mixing time of

Θ(n2) and order n inverse gap [11, 52]) to apply here as predicted for the low-

temperature Ising model. Of course it is a central open problem to establish this

Θ(n2) scaling, and the mean-curvature flow picture even for the Ising model at any

low but non-zero temperature (c.f., important progress at zero-temperature [17]).

Since this proposed picture drives our results in Chapters 7–8 for critical FK

dynamics, we close with the following broader, open-ended question.

Question 1.5.6. How closely does the behavior of the dynamical low-temperature

Ising model with plus boundary conditions translate to the critical FK model with

q > 4 and, say, free boundary conditions? What about to the Swendsen–Wang

dynamics of the critical Potts model with q > 4 and free boundary conditions?

31

Chapter 2

Preliminaries

In this chapter, we review the definitions of the Potts and random-cluster prop-

erties, discuss important properties of their phase transition in Z2, and introduce

some of the tools that will be central to our analysis throughout this thesis. More

specialized tools and techniques (e.g., cluster expansion) which will only appear

in specific chapters are defined locally therein. In Section 2.1, we describe the

Potts and random-cluster models on subsets of Z2, define the relevant notation,

and describe the rich phase transition they undergo at the critical βc(q) and pc(q)

respectively. In Section 2.2, we overview key definitions from the theory of Markov

chain mixing times, define the natural Markov chains for the Potts and random-

cluster models, and describe the main dynamical tools we will appeal to, in order

to obtain the desired bounds on their mixing time and spectral gap.

2.1 The Potts and FK models

We begin by redefining the Potts and random-cluster (FK) models and describing

their phase transition in Z2. For a more detailed survey of the random-cluster

32

model, and many of the key ideas described in this section, refer to [47].

2.1.1 Potts model

The (ferromagnetic) q-state Potts model on a graph G = (V,E) is the probability

distribution over configurations σ ∈ Ωp = [q]V (viewed as assignments of colors

out of [q] = 1, ..., q to the vertices of G) in which the probability of σ w.r.t. the

inverse-temperature β > 0 and the boundary conditions ζ (an assignment of colors

in [q] to the vertices of some boundary subset H ⊂ V , i.e. ζ ∈ [q]H) is given by

µζG,β,q(σ) =1

Zp

1σH = ζ exp(− β

∑u∼v

1σ(u) 6= σ(v)),

where the sum is over unordered pairs of adjacent vertices (u, v) ∈ E, σH indicates

the restriction of σ to H, and the normalizing constant Zp is the partition function.

In this thesis, we focus on graphs that are rectangular subsets of Z2 with nearest

neighbor edges ((u, v) ∈ E if |u− v| = 1) and vertex set

Λn,m := J0, nK× J0,mK = k ∈ Z : 0 ≤ k ≤ n × k ∈ Z : 0 ≤ k ≤ m ,

so that the notation Ja, bK stands for k ∈ Z : a ≤ k ≤ b. We use the abbreviated

form Λn = Λn,n and just Λ when both n and m are made clear from the context.

For general subsets S ⊂ Z2, the edge-set E(S) is the set of all edges of Z2 having

both ends in S; the boundary vertex set ∂S will be the set of vertices in S with a

neighbor in Z2 − S; we set the interior So = S − ∂S with edge set E(S)− E(∂S).

When considering rectangles Λ, denote the northern (top) boundary of Λ by

∂sΛ := J0, nK× n, define ∂s, ∂w and ∂e analogously, and let multiple subscripts

33

denote their union, i.e., ∂e,wΛ = ∂eΛ ∪ ∂wΛ.

2.1.2 Random-cluster (FK) models

For a graph G = (V,E), a random-cluster (FK) configuration ω ∈ Ωrc = 0, 1E

assigns binary values to the edges of G, either open (1) or closed (0). (In the

context of boundary conditions, these are often referred to instead as wired and free,

respectively). A cluster is a maximal connected subset of vertices that are connected

by open bonds, where singletons count as individual clusters. Every configuration

ω is naturally identified with a subset of E given by e ∈ E : ω(e) = 1.

For a boundary subset H ⊂ V , an FK boundary condition ξ is a partition of

the vertices in H. For a boundary subset H ⊂ V and an edge subset F ⊂ E, we

say that a configuration ωF on F induces a boundary condition ξω if for every

v, w ∈ H, they are in the same element of the partition ξω if and only if they

are connected by a set of open edges in ωF (i.e., v and w are wired in ξω). We

sometimes use ωH to indicate the boundary partition induced by ω on the vertices

of H, and if v, w are wired in a boundary condition ξ, we write vξ←→ w.

In the specific case where S ⊂ Z2, a boundary condition ξ on ∂S is said to

be realizable if there exists an FK configuration ω on E(Z2)− E(S) such that ω

induces ξ: i.e., u, v ∈ ∂S are in the same connected component of ω if and only if

they are wired in ξ.

Remark 2.1.1. Realizable boundary conditions are the most natural class of

boundary conditions (see [47]) since they are compatible with the geometry (i.e.

planarity) of Z2. However, non-realizable boundary conditions are still relevant

in some cases; for example, when considering the random-cluster model on non-

lattice graphs such as trees. In Section 6.5, we consider the mixing time of the FK

34

dynamics under non-realizable boundary conditions.

For p ∈ (0, 1) and q > 0, the FK model on G with a boundary condition ξ on

H ⊂ V is the probability measure over configurations ω ∈ 0, 1E given by

πξG,p,q(ω) =1

Zrc

po(ω)(1− p)|E|−o(ω)qk(ω;ξ),

where o(ω) is the number of open edges in ω and k(ω; ξ) counts the number of

connected components in the augmented graph (V, ωξ) and ωξ adds auxiliary edges

between all pairs of vertices in H that are in the same element of ξ. The partition

function Zrc is again the proper normalizing constant.

We sometimes interchange vertex sets with the subgraph they induce; e.g., the

random-cluster configuration on a set R ⊂ Z2 corresponds to the random-cluster

configuration in the subgraph induced by (R,E(R)). We omit the subscripts p, q

when understood from context.

Edwards–Sokal Coupling

The Edwards–Sokal coupling [35] provides a way to move back and forth

between the Potts model and the random-cluster model on a given graph G for

q ∈ 2, 3, . . .. The joint probability assigned by this coupling to (σ, ω), where

σ ∈ Ωp is a q-state Potts configuration at inverse-temperature β > 0 and ω ∈ Ωrc

is an FK configuration with parameters (p = 1− e−β, q), is proportional to

∏(xy)∈E(G)

[(1− p)1ω(xy) = 0+ p1ω(xy) = 1, σ(x) = σ(y)

].

35

It follows that, starting from a Potts configuration σ ∼ µG,β,q, one can sample an FK

configuration ω ∼ πG,p,q by letting ω(e) = 1 (e ∈ ω) with probability p = 1− e−β if

the endpoints x, y of the edge e have σ(x) = σ(y), and ω(e) = 0 (e /∈ ω) otherwise.

Conversely, from ω ∼ πG,p,q, one obtains σ ∼ µG,β,q by assigning an i.i.d. color in [q]

to each cluster of ω (i.e., σ(x) assumes that color for every vertex x of that cluster).

In the presence of boundary conditions ζ ∈ [q]∂G for the Potts model, it is

possible to sample σ ∼ µζG,β,q using the random-cluster model as follows. Associate

to ζ the FK boundary conditions ξ that wire x, y ∈ ∂G to each other if and only

if ζx = ζy, i.e., the partition corresponding to ξ is given by the color-classes of ζ.

Further denote by Fζ the random-cluster event that no boundary sites of different

colors under ζ are connected via ω: no two boundary sites x, y with ζ(x) 6= ζ(y)

are connected via ω in G:

Fζ =ω ∈ 0, 1E(G) : x

ω←→ y =⇒ ζx = ζy, for all x, y ∈ ∂G.

Then one can sample a configuration of µζΛ,β,q by first sampling ω ∼ πξΛ,p,q(· | Fζ)

for p = 1− e−β, then coloring the boundary clusters as specified by ζ, and coloring

every other cluster by an i.i.d. color uniformly over [q]. For further details, see [64],

where sampling under Fζ was described in the context of the FK-Ising model.

Monotonicity and the FKG inequality

While the Potts model for q 6= 2 does not obey any monotonicity property, the

FK model indeed does as long as q ≥ 1; this is central to the analysis of the Potts

model via its FK representation, and plays a crucial role in most of the proofs

throughout this thesis. The absence of this monotonicity is also the central obstacle

36

to a better understanding of the FK model when q ∈ (0, 1).

Let ω ≤ ω′ in the natural partial order on configurations, i.e., if ω(e) ≤ ω′(e)

for every e ∈ E. A function is called increasing if it is non-decreasing in this partial

order, and an event is called increasing if its indicator function is increasing (so

that the event is closed under addition of (open) edges). For every q ≥ 1 and every

p ∈ [0, 1], the model enjoys the FKG inequality [37]:

EπξG,p,q[f(ω)g(ω)

]≥ EπξG,p,q [f(ω)]EπξG,p,q [g(ω)] for every increasing f, g ,

and in particular, taking f and g to be indicator functions,

πξG,p,q(A ∩B) ≥ πξG,p,q(A)πξG,p,q(B) for every increasing events A,B .

This FKG inequality yields a monotonicity in boundary conditions for the FK

model. Define a partial order over boundary conditions by ξ ≤ η if the partition

corresponding to ξ is finer than that of η. The extremal boundary conditions

then, are the free boundary where ξ = v : v ∈ ∂G, which we denote by ξ = 0,

and the wired boundary where ξ = ∂G, denoted by ξ = 1. When q ≥ 1, the

FK model satisfies the following monotonicity in boundary conditions: if ξ, η are

two boundary conditions on ∂G with ξ ≤ η, then πξG,p,q πηG,p,q, where denotes

stochastic domination.

37

The Domain Markov property

The domain Markov property of the Potts model states that on a graph G =

(V,E), for every subset S ⊂ V and every configuration η

µ(σS−∂S ∈ · | σ(V−S)∪∂S = η) = µη∂SS

where, throughout the thesis, σS = σS = σ(S) denote the restriction of a configu-

ration σ to S. The Domain Markov property of the FK model states that, on any

graph G = (V,E), for every subgraph (S,E(S)) and every FK configuration ζ,

π(ωE(S) ∈ · | ωE−E(S) = ζ

)= πζS ,

where the boundary conditions ζ is the partition induced by the configuration ζ on

the vertices of ∂S.

Planar duality

Let Λ∗n = (Λ∗n, E(Λ∗n)) denote the planar dual of Λn. That is, Λ∗n corresponds

to the set of faces of Λn, and for each e ∈ E(Λn), there is a dual edge e∗ ∈ E(Λ∗n)

connecting the two faces bordering e. The FK distribution satisfies πΛn,p,q(ω) =

πΛ∗n,p∗,q(ω

∗), where ω∗ is the dual configuration to ω given as follows: for every

primal edge e and its dual-edge e∗ (intersecting at their midpoints) ω∗(e∗) = 1 if

and only if ω(e) = 0, and where

p∗ =q(1− p)

q(1− p) + p.

38

Under a realizable boundary condition ξ, this distributional equality becomes

πξΛn,p,qd= πξ

∗

Λ∗n,p∗,q∗ , where ξ∗ is the boundary condition induced by taking the dual

configuration of the configuration on E(Z2)− E(Λn) identified with ξ. Notice that

Z2 is isomorphic to its dual. The unique value of p satisfying p = p∗, denoted psd(q),

is called the self-dual point.

Infinite-volume Gibbs measures

Extra care is needed to define the Potts and FK models on infinite graphs,

as their partition functions become infinite. Infinite-volume Gibbs measures are

therefore defined via what is known as the DLR conditions : for an infinite graph

G = (V,E), a Potts measure µG,β,q on [q]V , defined in terms of its finite-dimensional

distributions, satisfies the DLR conditions if for every finite subset S ⊂ V ,

EµG,β,q(σG\S∈·)[µG,β,q

(σS ∈ · | σG\S

)]= µG,β,q(σS ∈ ·) .

An infinite-volume FK measure is defined analogously.

On infinite lattices, e.g., Z2, infinite-volume Gibbs measures can arise as weak

limits of finite-volume measures, say n → ∞ limits of the Potts/FK models on

Λn with certain prescribed boundary conditions. In particular, exploiting the

monotonicity in boundary conditions, all possible infinite-volume FK measures

lie in between (stochastically) the extremal ones obtained by taking wired and

free boundary conditions on Λn: we denote these limiting distributions by π1Z2,p,q

and π0Z2,p,q, and note that they may or may not be distinct. The uniqueness/non-

uniqueness of infinite-volume Gibbs measures is a canonical way by which the Potts

phase transition is characterized.

39

Potts and FK phase transitions

The FK model on Z2 exhibits a phase transition characterized by the emergence

of an infinite connected component. That is, there exists a critical value p = pc(q)

such that if p < pc(q), then, almost surely, all connected components of ω ∼ π1Z2,p,q

are finite, and if p > pc(q) then ω ∼ π0Z2,p,q almost surely has an infinite component.

For q ≥ 1, the exact value of pc(q) for Z2 was recently settled in [5], proving

pc(q) = psd(q) =

√q

√q + 1

,

(previously known only at q = 1 and q = 2 due to [54] and [56, 75]).

In particular, the work [5] implied that for q ≥ 1, for every p 6= pc(q), there is

a unique infinite-volume Gibbs measure so that πZ2,p,q := π1Z2,p,q = π0

Z2,p,q. By the

Edwards–Sokal coupling, the implications of this picture for the infinite-volume

measures of the Potts model are the following: when β < βc(q), the Potts model

has a unique infinite-volume measure, whereas when β > βc(q), it has multiple,

with infinite-volume measures consisting of mixtures of q extremal measures corre-

sponding to situations in which the FK infinite cluster gets each of the q possible

colors.

For two vertices x, y ∈ V , denote by x ←→ y the event that x and y belong

to the same connected component of ω, and for subsets S,R ⊂ V , denote by

S ←→ R the event that there exist x ∈ S, y ∈ R such that x ←→ y. With this

notation in hand, the FK phase transition is sharp in the sense that there is a

strong exponential decay of connectivities (which we sometimes refer to as the EDC

property). Namely, a consequence of the results in [2, 5] is that for every q > 1 and

p < pc(q), there is a c = c(p, q) > 0 such that for every boundary condition ξ and

40

all u, v ∈ Λn,

πξΛn,p,q(uΛn←→ v) ≤ e−cd(u,v) , (2.1.1)

where d(u, v) is the graph distance between u, v in Z2 and uΛn←→ v denotes that

there is an open path between u and v in the FK configuration on E(Λn) (not

using the connections of ξ). By the planar duality on Z2, when p > pc(q), there is

an analogous exponential decay of dual-connections.

At its critical point p = pc(q), we say the FK phase transition is continuous

or second-order if there exists a unique infinite-volume Gibbs measure (having no

infinite cluster), and discontinuous or first-order if there exist multiple infinite-

volume Gibbs measures (and under π1Z2,p,q there is almost surely an infinite cluster).

Our results at criticality hinge on the results of [34] characterizing the behavior

of the FK model at p = pc(q), summarized as follows, and sometimes referred to as

a dichotomy theorem.

Theorem 2.1.2 ([34, Theorem 3]). Let q ≥ 1; the following statements for the

critical FK model on Z2 are equivalent:

1. Discontinuous phase transition: π0Z2,pc,q

6= π1Z2,pc,q

.

2. Exponential decay of correlations under π0Z2,p,q: there exists c(p, q) > 0 so that

π0Z2,pc,q

((0, 0)←→ ∂J−n, nK2

)≤ e−cn . (2.1.2)

As mentioned, prior to the breakthrough work of [34], continuity of the phase

transition was known at q = 1 due to the seminal work of [54] and at q = 2

by Onsager, who also established that at p = pc(q) and q = 2, correlations

41

decay polynomially with critical exponent 14. Then along with the above theorem,

Duminil-Copin, Sidoravicius and Tassion [34] proved that the phase transition

is continuous at every q ∈ (1, 4]; this is believed to be the case for all q ∈ (0, 4]

but the absence of FKG inequalities leaves this as a major open problem in

the window q ∈ (0, 1) [4, 47]. Their proof combined the machinery of discrete

holomorphic observables introduced by Smirnov [88] with Russo–Seymour–Welsh

(RSW) estimates on crossing probabilities; this latter tool will be used extensively

in our proofs of Theorems 1 and 3 and is described in detail in Chapter 3.

Discontinuity of the phase transition, conjectured for all q > 4, was first

proved by Kotecky and Shlosman [55] for sufficiently large q using a perturbative

approach known as cluster expansion; the proof in [58] applies whenever q1/4 >

(κ+√κ2 − 4)/2, where κ is the connective constant of Z2. Plugging in the rigorous

bound κ < 2.6792 due to [79] affirms the phase coexistence for all q > 24.78. More

recently, by using relations between the FK model and the six-vertex model, [32]

verified that indeed the phase transition is discontinuous for all q > 4. As a

consequence, there are two extremal infinite-volume FK measures π1Z2,pc,q

6= π0Z2,pc,q

corresponding to the wired and free phases: this induces q + 1 distinct extremal

Potts measures on Z2, with q corresponding to the low-temperature phases where

a single color is dominant, along with a q + 1’th disordered phase.

2.2 Markov chain mixing times

In this section, we describe some of the central ideas from the theory of Markov

chain mixing times. We then define the (heat-bath) Glauber dynamics for the Potts

and random-cluster models, as well as the Swendsen–Wang dynamics, and present

42

the central techniques of block dynamics, censoring, and canonical paths to the

analysis of dynamics for spin-systems. For more details on Markov chain mixing

times and Glauber dynamics see [60] and [67], respectively.

Consider a discrete-time Markov chain (Xt)t≥0 with finite state space Ω, transi-

tion kernel P and stationary distribution π. In the continuous-time setting, instead

of P t consider the heat kernel given by

Ht(x, y) = Px(Xt = y) = etL(x, y) ,

where Px(Xt = y) = P(Xt = y | X0 = x) and L = limt↓01t(Ht − I) is the generator.

(We will define many of the relevant quantities (e.g., coupling distance, mixing

time) in discrete time, with the continuous-time versions obtained by replacing P t

by Ht and the transition probabilities P (x, y) by the rates c(x, y).)

Spectral gap

The mixing time of the Markov chain is closely related to the gap in its spectrum:

in discrete-time, gap := 1 − λ2 where λ2 is the second largest eigenvalue of P ;

in continuous-time it is the smallest non-zero eigenvalue in the spectrum of the

generator L. An important variational characterization of the gap is given by

gap = inff∈L2(π)

Varπ(f)>0

E(f, f)

Varπf, (2.2.1)

where the Dirichlet form E(f, g) is defined by

E(f, g) =1

2

∑x,y∈Ω

π(x)P (x, y)(f(y)− g(x))2 , (2.2.2)

43

for a discrete-time Markov chain, and by 〈−Lf, g〉L2(π) for a continuous-time process.

Mixing times

Denote the (worst-case) total variation distance between Xt and π by

dtv(t) = maxx∈Ω‖P t(x, ·)− π‖tv ,

where the total variation distance between two probability measures ν, π on Ω is

‖π − ν‖tv = supA⊂Ω

[π(A)− ν(A)] = 12‖π − ν‖L1 .

Further define the coupling distance

dtv(t) = maxx,y∈Ω

‖P t(x, ·)− P t(y, ·)‖tv ,

noting that dtv is submultiplicative and dtv(t) ≤ dtv(t) ≤ 2dtv(t). The total

variation mixing time of the Markov chain with respect to the precision parameter

0 < δ < 1 is

tmix(δ) = inftt : max

x∈Ω‖P t(x, ·)− π‖tv < δ .

For any choice of δ < 12, the quantity tmix(δ) enjoys submultiplicativity thanks

to the aforementioned connection with dtv; we write tmix, omitting the precision

parameter δ, to refer to the standard choice of δ = 1/(2e).

Another useful characterization of the total-variation distance is as

‖π − ν‖tv = inf(X,Y )∼(π,ν)

P(X 6= Y )

44

where the infimum is over all couplings Pr of the two measures π and ν. A (one step)

coupling of the Markov chain M specifies, for every pair of states (Xt, Yt) ∈ Ω× Ω,

a probability distribution over (Xt+1, Yt+1) such that the processes Xt and Yt,

viewed in isolation, are faithful copies ofM, and if Xt = Yt then Xt+1 = Yt+1. The

coupling time, denoted Tcoup, is the minimum T such that P(XT 6= YT ) ≤ 1/(2e),

starting from the worst possible pair of configurations X0, Y0. The following

inequality is standard : tmix ≤ Tcoup.

Relation between inverse spectral gap and mixing time

The total variation mixing time is bounded from below and from above via

the spectral gap, up to polynomial factors in the number of vertices. Let gap? be

the absolute spectral gap, which, for a discrete-time chain, is defined as gap? =

mini(1− |λi|) where the λi’s are the nontrivial eigenvalues of the transition kernel.

In our applications, as the spectrum of P or L is positive, gap? = gap. Then, for

any aperiodic, reversible Markov chain P with stationary measure π,

gap−1 − 1 ≤ tmix ≤ gap−1? log(2e · π−1

min) , (2.2.3)

where πmin := minxπ(x) (see, e.g., [60, §12.2]). For the FK and Potts models on a

box with O(n2) vertices and fixed 0 < p < 1 and q ≥ 1, there exists c(p, q) > 0 such

that πmin & e−cn2, from which it follows that tmix are gap−1 are equivalent up to

nO(1)-factors. As our theorems are primarily at polynomial in n levels of precision,

this comprarison (2.2.3) justifies our moving back and forth between spectral gap

and mixing time.

45

2.2.1 Dynamics for Potts and FK models

Heat-bath Potts Glauber dynamics

The discrete-time heat-bath Glauber dynamics for the Potts model onG = (V,E)

with boundary conditions η is the following reversible Markov chain w.r.t. µG,β,q:

given a configuration σ(t), one generates the configuration σ(t+ 1) by selecting a

vertex v in V uniformly at random, and (1) letting σw(t+ 1) = σw(t) for all w 6= v

and (2) sampling σv(t+ 1) according to µηG,β,q(σv ∈ · | (σw(t))w 6=v) . In particular,

the probability that the new color assigned to v will be k ∈ [q] is proportional to

exp(β∑

w∼v 1σw(t) = k).

The continuous-time heat-bath Glauber dynamics for the Potts model assigns

i.i.d. rate-1 Poisson clocks to all vertices in V ; when a clock at a site v rings, the

chain resamples σv according to the same rates described above.

Heat-bath FK Glauber dynamics

The discrete-time heat-bath Glauber dynamics for the FK model on G = (V,E)

with boundary conditions ξ is the following reversible Markov chain w.r.t. πG,p,q.

Given a configuration ω(t), one generates ω(t + 1) by selecting an edge e ∈ E

uniformly at random, and (1) letting ωf(t + 1) = ωf(t) for all f 6= e and (2)

sampling ωe(t+ 1) according to Bernoulli(p) if x←→ y in the configuration ω(t)

restricted to E − e (counting the connections induced by ξ), and according

to Bernoulli( pp+q(1−p)) otherwise. Its continuous-time version is defined by again

assigning each edge an i.i.d. rate-1 Poisson clock, and when the clock at an edge e

rings, resampling it as in item (2).

46

FK Glauber dynamics and planar duality

Each run of the FK-dynamics on Λn, with realizable boundary conditions ξ

and parameters p, q, determines a valid run of the FK-dynamics on the dual graph

Λ∗n with boundary conditions ξ∗ and parameters p∗, q. (Simply identify the FK

configuration in each step with its dual configuration; it can be straightforwardly

verified that the transitions of the FK-dynamics on the dual graph occur with the

correct probabilities.) Hence, the two dynamics have the same mixing times.

Remark 2.2.1. The edge-set of the dual graph Λ∗n is not exactly in correspondence

with the edge-set of a rectangle Λ∗ = −12, ..., n+ 1

2 × −1

2, ..., n+ 1

2 as it does

not include any edges that are between boundary vertices of Λ∗. All the proofs in

the paper carry through, only with the natural minor geometric modifications, to

the case of rectangles Λn with modified edge-set that only contains edges edges

with at least one endpoint in Λn \ ∂Λn. The dual of this modified graph is then a

(n−1)×(n−1) rectangle with all nearest-neighbor edges. With these considerations,

it often suffices for us to prove our theorems for p < pc(q). For example, the dual of

a realizable boundary condition is also realizable (induced by the dual configuration

on Z2 \ Λn), so it is sufficient to prove Theorem 5 for p < pc(q).

Notice that the definition of realizability of the boundary condition is slightly

different depending on the above choice of edge-set for a rectangle (while ξ is

encoded in a configuration on E(Z2) \ E(Λn), its dual ξ∗ would be encoded in a

configuration E((Z2)∗) \ (E(Λ∗) \ E(∂Λ∗))). However, it is easy to see that these

two notions of realizability are equivalent: a partition of ∂Λn can be encoded in a

configuration on E(Z2) \ E(Λn) if and only if it can be encoded in a configuration

on E(Z2) \ (E(Λn) \ E(∂Λn))

47

Cluster dynamics

Swendsen–Wang dynamics for the q-state Potts model on G = (V,E) at inverse-

temperature β is the following discrete-time Markov chain reversible w.r.t. µG,β,q.

From a spin configuration σ ∈ Ωp on G, generate a new state σ′ ∈ Ωp as follows.

1. Introduce auxiliary FK edge variables (ωe)e∈E as follows: for e = xy, set

ωe = 0 if σx 6= σy and ωe ∼ Bernoulli(p) if σx = σy.

2. For every connected component C of the graph (V, e : ωe = 1), assign an

i.i.d. color ΞC uniform among [q], and set σ′x = ΞC for every x ∈ C.

In the presence of boundary conditions, step (2) of the Swendsen–Wang dynamics

does not reassign the color of any cluster that contains a boundary vertex; instead

for those clusters, the color assignment is dictated by the boundary condition.

Remark 2.2.2. The definition of the Swendsen–Wang dynamics indicates why

it has fast mixing at low-temperature whereas the Potts Glauber dynamics slows

down on (Z/nZ)2. When β > βc, the metastable states are the q ordered phases,

corresponding to the q colors; but step (2) above recolors all FK clusters, and in

particular, reassigns the large macroscopic cluster of the ordered phase an i.i.d. color,

allowing the dynamics to easily jump between the q metastable states. However, as

it relies heavily on the FK representation of the model, the order-disorder energy

barrier may still be hard to overcome when both ordered and disordered phases are

metastable, e.g., when π1Z2 6= π0

Z2 .

Chayes–Machta dynamics for the FK model on G = (V,E) with parameters

(p, q), for q ≥ 1 and 0 < p < 1, is the following analogous discrete-time reversible

Markov chain: From an FK configuration ω ∈ Ωrc on G, generate a new state

ω′ ∈ Ωrc as follows.

48

1. Assign each cluster C of ω an auxiliary i.i.d. variable XC ∼ Bernoulli(1/q).

2. Resample every e = xy such that x and y belong to clusters with XC = 1 via

i.i.d. random variables Xe ∼ Bernoulli(p), to obtain the new configuration ω′.

In the presence of boundary conditions, step (1) counts clusters of ω by including

the auxiliary connections dictated by the boundary conditions.

Variants of Chayes–Machta dynamics with 1 ≤ k ≤ bqc “active colors” have

also been studied, with numerical evidence for k = bqc being the most efficient

choice; see [39].

Spectral gap comparisons

The following comparison inequalities between the above Markov chains are

due to Ullrich (see [91, Thm. 1], [92, Thm 4.8 and Lem. 2.7]).

Theorem 2.2.3 ([91, 92]). Let q ≥ 2 be integer. Let gapp and gaprc be the spectral

gaps of discrete-time Glauber dynamics for the Potts and FK models, respectively,

on a graph G = (V,E) with maximum degree ∆ and no boundary conditions, and

let gapsw be the spectral gap of Swendsen–Wang. Then we have

gapp ≤ 2q2(qe2β)4∆gapsw , (2.2.4)

(1− p+ p/q)gaprc ≤ gapsw ≤ 8gaprc |E| log |E| . (2.2.5)

The proof of (1.2.2) further extends to all real q > 1, whence

gaprc . gapcm . gaprc |E| log |E| , (2.2.6)

as was observed (and further generalized) by Blanca and Sinclair [10, §5], where

49

gapcm is the spectral gap of Chayes–Machta dynamics, and the constants in the

comparisons . are independent of ∆ and depend only on p and q. In particular,

because the estimates of Theorem 1.2.2 hold for general graphs G and are formulated

in the absence of boundary conditions, they hold immediately for Λ with free

boundary conditions, and with periodic boundary conditions (where it is identified

with the model on (Z/nZ)2 with no boundary conditions).

Potts boundary conditions and spectral gap comparisons

The comparison estimates in Theorem 1.2.2 are only valid in the absence of

boundary conditions, whereas much of this thesis considers boundary conditions

other than periodic and free. For other boundary conditions, we can deform Λn as

in the following remark; this could, however, distort the maximum degree ∆ by an

order n factor, leading to an exponential in n cost in (1.2.1).

Remark 2.2.4. For any FK boundary condition ξ on G, we can define a (not

necessarily planar) graph G by identifying all vertices of each boundary component

of ξ with a single vertex in G, and keeping the same edge structure. Then the FK

Glauber dynamics on G with no boundary conditions is the same as that on G

with boundary conditions ξ. In such a case, let ∂G denote the set of vertices in G

that arise from the boundary components of G.

Remark 2.2.5. The exponential dependence on ∆ in Eq. (1.2.1) can be improved

to exponential in the maximum degree of all but one vertex (see [91, Theorem 1’]),

whence ∆ in Eq. (1.2.1) can be replaced with the second largest degree of G. This

implies that Theorem 9 in fact also holds for the Potts Glauber dynamics.

The following is a consequence of the spin-flip symmetry of the Swendsen–Wang

dynamics.

50

Fact 2.2.6. Consider Swendsen–Wang dynamics on G with Potts boundary con-

ditions η, by considering the graph G, where boundary vertices of each color are

identified as single vertices. Let gapsw(Gη) be the spectral gap of Swendsen–Wang

dynamics on G with b.c., η, and let gapsw(Gη) be the spectral gap of Swendsen–

Wang dynamics on G with ∂G assigned the colors given by η. If ∂G consists of at

most one vertex, then

gapsw(Gη) = gapsw(Gη) = gapsw(G) .

Remark 2.2.5 and Fact 2.2.6 imply that for Λn with FK boundary conditions ξ

with at most one nontrivial boundary component, corresponding to Potts b.c., η

(an assignment of a color, e.g., red, to the nontrivial component of ξ and no other

color assignments),

gaprc . gapsw . n2 log n · gaprc .

Notice that all of the boundary conditions considered in Theorems 9–11 are of this

form. Thus, it suffices to prove Theorems 9–11 for the FK Glauber dynamics and

the implication for Swendsen–Wang dynamics follows immediately.

2.2.2 Dynamical tools

Monotonicity and the grand coupling

A discrete-time Markov chain with state space Ω and transition kernel P is

monotone if µP νP for every two probability distributions µ, ν on Ω such that

µ ν. In continuous-time, this translates to Ht(µ, ·) Ht(ν, ·) for every µ ν.

The random mapping representation of the discrete-time FK Glauber dynamics

51

on a graph G = (V,E) views the updates as a sequence (Ji, Ui)i≥1, in which Ji’s are

i.i.d. uniform edges (the updated locations), and the Ui’s are i.i.d. uniform on [0, 1]:

starting from an initial configuration ω0, at time i, writing Ji = (x, y), the dynamics

replaces the value of ω(Ji) by 1Ui ≤ p if x←→ y in the configuration on E−Ji

(in the presence of boundary conditions, including boundary connections) and by

1Ui ≤ pp+q(1−p) otherwise. The grand coupling for FK Glauber dynamics is a

coupling of the chains from all initial configurations on G which, via the above

random mapping representation, uses the same update sequence (Ji, Ui)i≥1 for each

one of these chains. (In continuous time, the random mapping representation would

also include the sequence of update times (Ti)i≥1.) Using this representation and

the FKG inequality, one sees that heat-bath Glauber dynamics for the FK model

at q ≥ 1 is monotone.

Thus, for q ≥ 1, this coupling preserves the partial ordering of the Markov chains

started from all possible initial configurations, at all times t ≥ 0. In particular,

under the grand coupling, the value of an edge e in Glauber dynamics at time t

from an arbitrary initial state ω0, is sandwiched between the corresponding values

from the free and wired initial states; thus, by a union bound over all edges,

maxx∈Ω‖P t(x, ·)− π‖tv ≤ max

x,y∈Ω‖P t(x, ·)− P t(y, ·)‖tv

≤ |E(Λ)| ‖P t(1, ·)− P t(0, ·)‖tv . (2.2.7)

Censoring inequality

Key to our proof will be the Peres–Winkler [76] censoring inequality for monotone

systems. While the theorem of [76] and its subsequent applications in e.g., [63, 70]

are stated for spin systems whose sites are the vertices of the underlying graph,

52

one can view the edges as the sites by considering the appropriate line graph; it is

then easy to verify that the FK Glauber dynamics satisfies the conditions of [76,

Theorem 1.1]. Further, while Theorem 1.1 of [76] is stated for the discrete-time

dynamics, its formulation in continuous-time follows from the same proof: see

also [70, Theorem 2.5].

Theorem 2.2.7 ([76]). Let µT be the law of continuous-time Glauber dynamics

at time T of a monotone system on Λ with invariant measure π, whose initial

distribution µ0 is such that µ0/π is increasing. Set 0 = t0 < t1 < . . . < tk = T for

some k, let (Bi)ki=1 be subsets of Λ, and let µT be the law at time T of the censored

dynamics, started at µ0, where only updates within Bi are kept in the time interval

[ti−1, ti). Then ‖µT − π‖tv ≤ ‖µT − π‖tv and µT µT ; moreover, µT/π and µT/π

are both increasing.

Block dynamics

A key ingredient in the proof of [64], as well as our proof of Theorem 1, is the

block dynamics technique due to Martinelli (see [67, §3]) for bounding the spectral

gap of the Glauber dynamics. SupposeB1, ..., Bk are such thatB1−∂B1, ..., Bk−∂Bk

covers Λ− ∂Λ. Then the block dynamics is the corresponding Glauber dynamics

that updates one block (instead of one site) at a time: each block is assigned a rate-1

Poisson clock; when the clock at Bi rings, resample the configuration on Bi − ∂Bi

according to µσBi where the boundary conditions σ are given by the configuration

restricted to Λ− (Bi − ∂Bi).

Theorem 2.2.8 ([67, Proposition 3.4]). Consider the continuous-time Glauber

dynamics for the Potts model on Λ with boundary condition ζ, which is reversible

w.r.t. the Gibbs distribution µζΛ. Let gapp(Λζ) and gapB(Λζ) respectively be the

53

spectral gaps of the single-site dynamics on Λ and block dynamics corresponding to

B1, . . . , Bk such that Bo1, ..., B

ok cover Λo. Then letting χ = supx∈Λ #i : Bi 3 x,

we obtain

gapp(Λζ) ≥ χ−1gapB(Λζ) min

i=1,..,kminϕ

gapp(Bϕi ) .

An analogue in the context of FK Glauber dynamics is described in §6.1.6.

Canonical paths

The following well-known geometric approach (see [28, 29, 53, 86] as well as [60,

Corollary 13.24]) serves as an effective method for obtaining an upper bound on

the inverse gap of a Markov chain, and will be used in our proof of Theorem 8.

Theorem 2.2.9. Let P be the transition kernel of a discrete-time Markov chain on

Ω, reversible w.r.t. π, and write Q(x, y) = π(x)P (x, y) for every x, y ∈ Ω. For each

(a, b) ∈ Ω2, assign a path ζ(a, b) = (x0 = a, . . . , xn = b) such that P (xi, xi+1) > 0 for

all i, and write |ζ(a, b)| = nab, identifying ζ(a, b) with (xi−1, xi) : i = 1, . . . , nab.

Then,

gap−1 ≤ maxx,y∈Ω

Q(x,y)>0

1

Q(x, y)

∑a,b∈Ω

(x,y)∈ζ(a,b)

nabπ(a)π(b). (2.2.8)

A very standard application of Theorem 2.2.9 (see e.g., [66] in the setting of

the Ising model) proves upper bounds on mixing times of spin systems in terms of

the cut-width of the underlying graph.

Lemma 2.2.10. Consider the Glauber dynamics for the q-state Potts model at

inverse temperature β on a rectangle Λn,l = J0, nK × J0, `K for 0 ≤ ` ≤ n, with

54

arbitrary boundary conditions η. There exists a constant c(β, q) > 0 such that

gapp(Ληn,l)−1 . ec` .

Remark 2.2.11. In the absence of boundary conditions, the same bound holds

for the FK models by making the natural modifications and observing that in

the FK setting, the probability of any single edge-flip is at least some c(p, q) > 0.

However, the introduction of boundary conditions, which can encode long-range

correlations, destroys the standard bound on the cut-width of the n× ` rectangle,

and prevents this from going through. In fact, we will see later that under general

FK boundary conditions, the bound of Lemma 2.2.10 does not hold. We are able to

prove a somewhat satisfactory exp[c` log n] analogue under arbitrary realizable FK

boundary conditions, by using a tree-like structure of planar boundary conditions

to recursively bound the cutwidth on n× ` graphs in Section 6.4. That bound may

be of independent interest.

In particular, Lemma 2.2.10 and Remark 2.2.11 directly imply upper bounds

that are exponential in n for the inverse gaps in Theorems 2, 4 and 5.3.1.

Boundary modifications

Due to the difficulties of dealing with problematic FK boundary conditions, it

will be helpful to modify boundary conditions, say to remove long-range boundary

connections, and perform the dynamical analysis with the modified boundary

conditions. Let ξ, ξ′ be a pair of FK boundary conditions on Λ with corresponding

55

FK Glauber dynamics mixing times tmix, t′mix; define

Mξ,ξ′ = ‖πξΛ/πξ′

Λ‖∞ ∨ ‖πξ′

Λ /πξΛ‖∞ .

It is well-known (see, e.g., [70, Lemma 2.8]) that for some c independent of n, ξ, ξ′,

gaprc(Λξ) ≤M3ξ,ξ′gaprc(Λξ′) and tmix ≤ cM3

ξ,ξ′|E(Λ)|t′mix (2.2.9)

(this follows from first bounding tmix via its spectral gap, then using the variational

characterization of the spectral gap: the Dirichlet form, expressed in terms of local

variances, gives a factor of M2ξ,ξ′ , and the variance produces another factor of Mξ,ξ′).

The Radon–Nikodym factor Mξ,ξ′ can easily be bounded by a distance between

ξ and ξ′.

Definition 2.2.12. For any pair of boundary conditions, ξ, ξ′ define the symmetric

distance function d(ξ, ξ′) as follows: if ξ′′ is the unique smallest (in the previously

defined partial ordering) boundary condition with ξ′′ ≥ ξ and ξ′′ ≥ ξ′, define

d(ξ, ξ′) = k(ξ)− k(ξ′′) + k(ξ′)− k(ξ′′), where k(ξ) is the number of elements in the

partition induced by ξ.

In particular, if ξ is a boundary condition on Λ and ξ′ is a refinement of ξ, then

ξ′ ≤ ξ and d(ξ, ξ′) = k(ξ′)− k(ξ).

For an initial configuration ω0, and boundary condition ξ, let

dξω0(t) = ‖Pξω0

(Xt ∈ ·)− πξΛ‖tv .

where here and throughout, for any Markov chain (Xt)t≥0, we abbreviate P(Xt ∈

· | X0 = ω0) with boundary conditions ξ, to Pξω0(Xt ∈ ·).

56

Lemma 2.2.13. Let ξ, ξ′ be a pair of boundary conditions on ∂Λ. Then,

Mξ,ξ′ ≤ q2d(ξ′,ξ) , (2.2.10)

Let P be a distribution over coupled pairs of boundary conditions (ξ, ξ′) such that

P-a.s., Mξ,ξ′ is at most some M∆. For some universal c > 0, if t′ = ct|E|−2M−4∆ ,

E[dξ1(t) ∨ dξ0(t)] ≤ e−M∆ + 8E[dξ′

1 (t′) ∨ dξ′

0 (t′)] . (2.2.11)

In particular, taking P to be a point mass on (ξ, ξ′), for every t > 0,

maxω0∈0,1

dξω0(t) ≤ 8 max

ω0∈0,1dξ′

ω0

(c|E(Λ)|−2q−8d(ξ′,ξ) t

)+ exp

(−q2d(ξ′,ξ)

). (2.2.12)

Proof. By definition of d(ξ, ξ′), for any configuration ω, the difference in the number

of clusters under boundary conditions ξ vs. ξ′, |k(ω; ξ)− k(ω; ξ′)| is at most d(ξ, ξ′).

By definition of the FK model, every additional cluster receives a weight of q.

In order to prove (2.2.11), begin with the observation that by (2.2.9),

E[dξ1(t) ∨ dξ0(t)] ≤ e−M∆ + P(tmix ≥ t/M∆) ≤ e−M∆ + P(t′mix ≥ |E|t′) .

For t′mix ≥ s, there must exist an ω0 such that dξ′ω0≥ 1/(2e). But by Eq. (2.2.7),

dξ′ω0

(s) ≤ 2|E(Λ)|(dξ′

1 (s) ∨ dξ′

0 (s)), which implies that

P(t′mix ≥ |E(Λ)|t′) ≤ P

(dξ′

1 (|E(Λ)|t′) ∨ dξ′

0 (|E(Λ)|t′) ≥ (4e|E(Λ)|)−1

).

57

As a consequence of Theorem 2.2.7, it is well-known (see [70, Corollary 2.7]) that

dξ′

1 (t) ∨ dξ′

0 (t) ≤(

4(dξ′

1 (t0) ∨ dξ′

0 (t0)))bt/t0c

,

and thus

P(dξ′

1 (|E(Λ)|t′) ∨ dξ′

0 (|E(Λ)|t′) ≥ (4e|E(Λ)|)−1)≤ 8E[dξ

′

1 (t′) ∨ dξ′

0 (t′)] .

Conductances and Cheeger’s inequality

A powerful geometric technique for proving lower bounds on mixing times is

constructing a set S ⊂ Ω that is a bottleneck for the Markov chain dynamics. For

a chain with transition kernel P (x, y) and stationary distribution π, let the edge

measure between A,B ⊂ Ω be

Q(A,B) =∑ω∈A

π(ω)∑ω′∈B

P (ω, ω′) , (2.2.13)

(see, e.g., [60, Chapter 7]). Then the conductance/Cheeger constant of Ω is

Φ? = minS⊂Ω

Φ(S) where Φ(S) =Q(S, Sc)

π(S)π(Sc). (2.2.14)

and the following relation between Φ and the spectral gap of the chain [60] holds:

2Φ ≥ gap ≥ Φ2/2 . (2.2.15)

58

Chapter 3

Equilibrium estimates at q ∈ [1, 4]

and RSW Theory

At a continuous phase transition point i.e., q ≤ 4 and p = pc(q), the equilibrium

behavior of the Ising, Potts and FK models are expected to have a beautiful,

conformally invariant structure with a scaling limit described by conformal field

theory (CFT). In particular, the models are expected to be characterized by their

algebraic critical exponents, convergence of interfaces to SLE’s, and convergence of

fields of local observables to Euclidean fields in CFT. Since the seminal works of

Schramm introducing SLE ([83] as well as e.g., [80, 84]) and Smirnov introducing

the framework of discrete holomorphic observables [87, 88], such a scaling limit has

been established in the case of the Ising model q = 2 e.g., [14, 22, 23, 40, 51, 88]

(as well as Bernoulli percolation q = 1 on the hexagonal lattice e.g., [15, 16, 87]).

One particular feature of this stunning picture is that the probability of, say,

an open FK cluster crossing an n × αn rectangle with bounded aspect ratio α

remains bounded away from zero and one as n → ∞ (indicative of criticality

59

from a renormalization perspective): this is known as a Russo–Seymour–Welsh

(RSW) bound. In this chapter, we use percolation techniques and RSW theory,

to prove necessary equilibrium estimates on crossing probabilities, arm-exponents,

and numbers of disjoint long-range crossings for the FK model at a continuous

phase transition point. In addition to being central to the dynamical analysis of

the Potts and FK models at p = pc(q) and q ∈ (1, 4] (namely Theorems 1 and 3),

many of these estimates are independently of interest; we point in particular, to

the bounds on the number of nested boundary components (bridges) in a typical

realizable boundary, derived in Section 3.2.

Throughout the chapter, take q ∈ (1, 4], let p = pc(q) and drop p, q from the

notation.

3.1 Crossing probabilities and RSW Theory

In this section we present estimates on crossing probabilities that will be used

to prove the desired mixing time bounds. Russo–Seymour–Welsh (RSW) bounds

were first introduced [81, 85] in the context of Bernoulli percolation, and are central

to the understanding of Bernoulli percolation at its critical point p = 12.

In [33], RSW estimates were proven for the FK representation of the Ising

model, q = 2, using the framework of discrete-holomorphic observables from [88];

in this setting, where the boundary conditions play a role, the bounds on crossing

probabilities held uniformly over the boundary conditions on the rectangle Λn,αn.

For 1 < q ≤ 4, the proof of continuity of the phase transition [34] relied crucially

on analogous RSW estimates, which we develop in detail in this section.

60

Crossing probability notation

Recall that for two vertices x, y ∈ V , we denote by x ←→ y the event that

x and y belong to the same cluster of ω. In the context of a subgraph S ⊂ G,

write xS←→ y to denote that x and y belong to the same cluster of ωE(S). For a

rectangular subset R ⊂ Z2, refer to the event

Cv(R) :=⋃

x∈∂sR , y∈∂nR

ω : x

R←→ y

as a vertical crossing of a rectangle R, and denote a horizontal crossing of the

rectangle R by

Ch(R) :=⋃

x∈∂eR , y∈∂wR

ω : x

R←→ y.

Consider a subset of Z2 of the form A = R2 −R1 where R1, R2 are rectangular

subsets of Z2 with R1 ( R2. Call such domains annuli, and define open circuits

as paths of nontrivial homology in A connecting a vertex x to itself. Denote the

existence of an open circuit in the annulus A by Co(A).

Finally, we add the ∗-symbol to the above crossing events to refer to the

analogous dual-crossings (configurations ω such that an open-crossing occurs in its

corresponding dual-configuration ω∗ on the appropriate dual subgraph).

Notice that crossing events e.g., x ←→ y, Ch(R) are increasing events, and

dual-crossing events are decreasing events.

61

RSW estimates

The following RSW estimates hold for the q ∈ [1, 4] FK model at p = pc(q)1

(note the difference between 1 < q < 4 and the extremal case q = 4, where full

RSW-type bounds up to the boundary are believed to fail [34]).

Theorem 3.1.1 ([34, Theorem 7]). Consider the critical FK model for 1 ≤ q < 4

on Λ = Λn,n′ with n′ = bαnc for fixed 0 < α ≤ 1. There exists p0 = p0(q, α) > 0

such that for every boundary condition ξ and all n,

πξΛ(Cv(Λ)) > p0 .

Theorem 3.1.2 ([34, Theorem 3]). Let q = 4 and consider the critical FK model

on Λ = Λn,n′ with n′ = bαnc for fixed 0 < α ≤ 1. Then for every ε, ε′ > 0 there

exists p0 = p0(α, ε, ε′) > 0 such that, for every boundary condition ξ and every n,

πξΛ(Cv(Jεn, (1− ε)nK× Jε′n′, (1− ε′)n′K) > p0 .

Theorem 3.1.3 ([34, Proposition 2]). Fix ε, ε′ > 0 and 0 < α ≤ 1, and consider

the critical FK model at 1 ≤ q ≤ 4 on the annulus A = Λn,n′ − Jεn, (1 − ε)nK ×

Jε′n′, (1 − ε′)n′K for n′ = bαnc. There exists p0 = p0(q, α, ε, ε′) so that, for every

boundary condition ξ and every n,

πξA (Co(A)) > p0 .

A consequence of the above RSW-type bounds is polynomial decay of correlations

1In [34], Theorems 3.1.2 and 3.1.3 were formulated for the special case of ε = ε′, but readilyextend to the more general setting presented here.

62

for the critical FK model at 1 ≤ q ≤ 4 (see, e.g., the proof of [34, Lemma 1]).

Theorem 3.1.4 (decay of correlations, [34]). For 1 ≤ q ≤ 4 and p = pc(q), there

exists a unique infinite-volume Gibbs measure πZ2, and there exist c1, c2 > 0 such

that

n−c1 . πZ2

((0, 0)←→ ∂J−n, nK2

). n−c2 .

It is believed that for every q ∈ [1, 4] the above decay of correlations behaves

as n−zq for some critical exponent zq > 0: this is known in the settings of q = 1

corresponding to critical percolation, and q = 2 corresponding to the Ising model.

In particular, this power-law decay at criticality suggests the conjectured conformal

invariance and fractal structure of the critical FK model on Z2 when q ≤ 4.

3.1.1 Crossing probabilities at q = 4

For 1 ≤ q < 4, the probability of a horizontal crossing of a rectangle with

arbitrary boundary conditions is uniformly bounded away from 0 (Theorem 3.1.1),

whereas at q = 4, under free boundary conditions, it is expected that the probability

of such a crossing of Λ in fact decays to 0 as n → ∞ [34]. We lower bound this

crossing probability under general boundary conditions.

Theorem 3.1.5. Let q = 4 and consider the critical FK model on Λ = Λn,n′, where

n′ = bαnc for a fixed aspect ratio 0 < α ≤ 1. There exist c(α), γ(α) > 0 such that

for every boundary condition ξ and every n,

πξΛ (Ch(Λ)) > cn−γ .

Proof. By monotonicity in boundary conditions, it suffices to prove the above

63

12n′

12n′ + 4α logn

12n′ + 8α logn

logn 2 logn 3 logn 5 logn 11 logn

R0

R1

R2

R3

R4

Figure 3.1: The first four steps of the stitching technique of Theorem 3.1.5. Re-peating log εn times creates a macroscopic horizontal open crossing.

for free boundary conditions (the case ξ = 0). Fix δ > 0 and let R = J0, nK ×

J(12− δ)n′, (1

2+ δ)n′K. We will show the stronger result that there exist some

γ(α), γ′(α) > 0 such that,

n−γ . π0Λ

((0, n′/2)

R←→ (n, n′/2)). n−γ

′. (3.1.1)

The upper bound in (3.1.1) is a consequence of the polynomial decay of correla-

tions in Theorem 3.1.4, and it remains to establish the lower bound. Observe that

for every e,

infω′∈0,1E(Λ)−e

πω′

Λ (e ∈ ω) =pc

pc + q(1− pc)=

1

1 +√q

;

thus, we can force all the edges of R0 = J0, 2 log nK×n′2 to be open with probability

at least (1 +√q)−2 logn = n−2βc , where we recall that βc = log(1 +

√q).

We boost this to a horizontal crossing of length δn/2 from the boundary by

stitching together horizontal and vertical crossings and applying the FKG inequality

(see Fig. 3.1). Fix ε > 0 sufficiently small (e.g., a choice of ε = δ/10 would suffice),

64

and consider

R2k−1 = J(2k − 1) log n, (3 · 2k−1 − 1) log nK× Jn′2, n′

2+ 2k+1α log nK ,

R2k = J(2k − 1) log n, (3 · 2k − 1) log nK× Jn′2, n′

2+ 2k+1α log nK

for k = 1, . . . , K, where K = blog2

(εn

logn

)c.

Moreover, take R2k−1 and R2k to be the concentric 32-dilations of R2k−1 and

R2k, respectively. By construction, each R2K−1 and R2K has width at most 2εn

and height at most 2εn′, hence their respective dilations R2K−1 and R2K are both

contained in R.

As a consequence of Ri ⊂ Λ the free boundary conditions on Ri are dominated

by the distribution over boundary conditions induced by π0Λ. Thus, there exists

some p1(α), p2(α) > 0 given by Theorem 3.1.2 such that

π0Λ(Cv(R2k−1)) ≥ π0

R1(Cv(R2k−1)) ≥ p1 , and likewise π0

Λ(Ch(R2k)) ≥ p2 .

(Notice the aspect ratios of R2k−1 are the same for all k, and similarly for R2k.)

Further, for every k, these events are increasing; thus by the FKG inequality,

π0Λ (Cv(R2k−1) ∩ Ch(R2k)) ≥ p1p2 .

At the final scale K, the width of R2K is (2− o(1))εn and its height is (2− o(1))εn′,

so

(ωR0

= 1 ∩K⋂k=1

(Cv(R2k−1) ∩ Ch(R2k))

)⊂ (0, n

′

2)

R←→ 2εn × J12n′, (1

2+ 2ε)n′K

65

for any sufficiently large n. By repeated application of the FKG inequality,

π0Λ

((0, n

′

2)

R←→ 2εn × J12n′, (1

2+ 2ε)n′K

)≥ n−2βc(p1p2)K = n−γ (3.1.2)

for some γ > 0. By symmetry, the exact same argument shows that

π0Λ

((n, n

′

2)

R←→ (1− 2ε)n × J12n′, (1

2+ 2ε)n′K

)≥ n−γ . (3.1.3)

In order to complete the desired horizontal crossing, we require an open path

connecting the left and right crossings, via an open circuit in the annulus A1 given

by

A1 = J0, nK× J(12− δ)n′, (1

2+ δ)n′K− Jεn, (1− ε)nK× J(1

2− 3ε)n′, (1

2+ 3ε)n′K .

By Theorem 3.1.3, there is an absolute constant p3(α) > 0 such that π0A1

(Co(A1)) >

p3. Since the induced boundary conditions on ∂A1 by π0Λ stochastically dominate

free boundary conditions on ∂A1, it follows that π0Λ (Co(A1)) > p3. Finally, the event

Co(A1) is increasing, and its intersection with the two horizontal crossing events

from (3.1.2) and (3.1.3) is a subset of the event (0, n′2

)R←→ (n, n

′

2). Thus, by

FKG, the latter has probability at least p3n−2γ , establishing (3.1.1), as desired.

The following is a slight extension of Theorem 3.1.5.

Proposition 3.1.6. Let q ∈ (1, 4] and fix α ∈ (0, 1]. Consider the critical FK

model on Λ = Λn,n′ with bαnc ≤ n′ ≤ dα−1ne. For every ε > 0, there exists

c?(α, ε, q) > 0 such that for every x ∈ Jεn, 1− εnK, and every boundary condition ξ

on ∂Λ, one has

πξΛ((x, 0)←→ (x, bn′c)

)& n−c? .

66

Proof. The proposition was proved in the case n′ = bαnc and x = n2. Since

the crossing probabilities of Theorem 3.1.2 are monotone in the aspect ratio,

each is bounded away from zero for aspect ratios in [α, α−1], yielding the desired

extension.

The next two results are for q = 4 (Theorem 3.1.1 implies both for 1 < q < 4).

Lemma 3.1.7. Let q = 4 and fix α ∈ (0, 1]. Consider the critical FK model on

Λ = Λn,n′ with bαnc ≤ n′ ≤ dα−1ne and (1, 0) boundary conditions denoting wired

on ∂sΛ and free elsewhere. For every ε > 0, there exists p = p(α, ε) > 0 such that

π1,0Λ (Cv(J0, nK× J0, (1− ε)n′K)) ≥ p .

Proof. Consider the rectangle Λ′ = J0, nK × J−εn′, n′K which contains Λ. By

monotonicity in boundary conditions and the domain Markov property, we see that

π1,0Λ (ω ∈ ·) sup

ξπ0

Λ′

(ωE(Λ) ∈ · | ωE(Λ′)−E(Λ) = ξ

) π0

Λ′(ωE(Λ) ∈ ·) .

By Theorem 3.1.2, there exists a p(ε, α) > 0 such that π0Λ′

(Cv(Λ)

)≥ p. Since Cv(Λ)

is an increasing event, these together imply the desired.

Corollary 3.1.8. Let q = 4 and fix α ∈ (0, 1]. Consider the critical FK model on

Λ = Λn,n′ with bαnc ≤ n′ ≤ dα−1ne and boundary conditions, denoted by (1, 0, 1, 0),

that are wired on ∂n,sΛ and free on ∂e,wΛ. Then there exists p(α) > 0 such that

π1,0,1,0Λ (Cv(Λ)) ≥ p .

Proof. For all n′ ≤ n this follows immediately from self-duality and monotonicity

67

in boundary conditions. For n ≤ n′ ≤ dα−1ne, by monotonicity in boundary

conditions and Lemma 3.1.7, for any ε ∈ (0, 1), there is a p(1, ε) > 0 such that,

π1,0,1,0Λ (Cv(J0, nK× J0, εnK)) ≥ p ,

and by reflection symmetry, π1,0,1,0Λ (Cv(J0, nK× Jn′ − εn, n′K)) ≥ p. Let

Aε = Λ− Jεn, (1− ε)nK× Jεn, n′ − εnK .

Since Co(Aε) can be lower bounded by four crossings of rectangles, each of whose

probabilities is monotone in the aspect ratio and thus bounded away from 0

uniformly over n ≤ n′ ≤ dα−1ne, we have that π1,0,1,0Λ (Cv(Aε)) ≥ p′ uniformly over

n ≤ n′ ≤ dα−1ne for some p′(α, ε). Now observe that

(Cv(J0, nK× J0, εnK) ∩ Cv(J0, nK× Jn′ − εn, n′K) ∩ Co(Aε)

)⊂ Cv(Λ) .

After fixing any small ε > 0, by the FKG inequality, there exists some p(α) > 0

such that for every n ≤ n′ ≤ dα−1ne, one has π1,0,1,0Λ (Cv(Λ)) ≥ p, as required.

3.1.2 Lower bounds on arm exponents

As mentioned, the polynomial decay is expected to have a critical exponent

predicted by the algebraic structure of conformal field theory; while this critical

exponent is known rigorously at q = 2 [75], this is not the case for other q ∈ (1, 4].

Here, we adapt a standard argument for obtaining the Bernoulli percolation two-arm

exponent, and use it to lower bound the FK one-arm exponent.

Lemma 3.1.9. Fix an ε > 0 and consider the critical FK model for q ∈ (1, 4] on

68

R = J−n, nK2; there exists c(q) > 0 such that

π0R

(0←→ ∂(J−n

2, n

2K2))≥ cn−

12 , (3.1.4)

and thus, there exists c′(ε, q) > 0 such that for every x, y ∈ J−(1− ε)n, (1− ε)nK2,

π0R

(x←→ y

)≥ c′|x− y|−1 . (3.1.5)

Proof. Let L = J−n8, n

8K× 0, for every x ∈ L, set Rx = x+ J−n

2, n

2K2, and let R+

x

be the top half of Rx. Also let B = J−3n4, 3n

4K2 and define the event

Sx = x←→ ∂nRx in R+x ∩ x+ (1

2, 0)

∗←→ ∂nRx in R+x .

Consider the event S that there exists a site x ∈ L such that Sx holds. We

begin by proving that π1B(S) ≥ c for some c > 0 independent of n. By using

Theorem 3.1.1–3.1.2 and stochastic domination twice, we see that there exists

c(q) > 0 such that

π1B(Cv(J−n

8,− n

10K× J0, n

2K) ∩ C∗v(J n10

, n8K× J0, n

2K)) ≥ c .

But one can observe that the above event implies that the right-most point on

L that is part of the cluster of the vertical open crossing in R+0 satisfies Sx, so

π1B(S) ≥ c. At the same time, we have by a union bound that

π1B(S) ≤

∑x∈L

π1B(Sx) ≤ (n/4) max

x∈Lπ1B(Sx) , so that

maxx∈L

π1B(Sx) ≥ 4cn−1 .

69

The maximum on the left-hand side is attained by some deterministic x ∈ L which

we set to be j, for which we have, by the FKG inequality and self-duality, that

π1B(Sj) ≤ π1

B

(j ←→ ∂nRj in R+

j

)π1B

(j + (1

2, 0)

∗←→ ∂nRj in R+j

)= π1

B

(j

R+j←→ ∂nRj

)π0B(j

R+j←→ ∂nRj) . (3.1.6)

By the RSW estimate, Theorem 3.1.3, we see that π0B(Co(B −

⋃j∈LRj)) ≥ ε for

some ε(q) > 0, and therefore by monotonicity in boundary conditions,

π0B

(j

R+j←→ ∂nRj

)≥ επ1

B

(j

R+j←→ ∂nRj

).

Plugging this in to (3.1.6) implies that π0B(j ←→ ∂Rj) ≥ 2

√cεn

. In order to com-

plete the proof of (3.1.4), we translate by −j to see that π0B−j(0←→ ∂J−n

2, n

2K2) ≥

c′n−12 for some c′(q) > 0. Since j ∈ L, B − j ⊂ R and by monotonicity, we

deduce (3.1.4).

Going from (3.1.4) to (3.1.5) is a standard exercise in using RSW estimates

(Theorems 3.1.1–3.1.2) and the stitching arguments used in the proof of Theo-

rem 3.1.5; since both x, y are macroscopically far from ∂R, we can use (3.1.4) to

connect each of them to some distance O(|x− y|) away, and stitch open crossings

to connect these two together via the FKG inequality and Theorems 3.1.1–3.1.2,

yielding the desired.

3.2 Boundary bridges

As discussed earlier, dual-crossings, whose probabilities have been bounded

uniformly in the boundary condition by Russo–Seymour–Welsh type estimates at

70

Figure 3.2: A pair of boundary connections (green, purple) constituting two distinctboundary bridges, separated by a (blue) dual connection.

q = 2 by [33] and at all q ∈ (1, 4] by [34], and considered in the preceding sections,

are central to coupling proofs of upper bounds on mixing times of Ising models

in [64] and Potts models in Chapter 4. When we move to the Glauber dynamics

for the FK model, there is a major new obstacle with which we have to deal. As

pointed out in the introduction (e.g., Figure 1.5), disjoint long-range FK clusters

along the boundary of rectangles, called bridges (see Fig. 3.2), prevent the coupling

described above and are an impediment to polynomial mixing time upper bounds

for the FK Glauber dynamics at q ∈ (1, 4] and p = pc(q) (see also Section 4.2.1).

Definition 3.2.1. Let Λn,m = J0, nK × J0,mK. Given an FK configuration ω on

E(Z2)− E(Λn,m) and a boundary edge e ∈ ∂Λn,m, say without loss of generality

e ∈ ∂nΛn,m, a bridge over e is an open FK cluster in ω that contains at least one

vertex in ∂nΛn,m to the left of e and one to the right. Let Be(ω) denote the set

of all bridges of e.

(For a more detailed definition, we also refer the reader to Section 3.2.1.) In

critical bond percolation (q = 1 and p = 1/2), the Berg–Kesten (BK) inequality

would suggest that π(|Be| > K log n + a) ≤ exp(−ca) for universal K, c > 0 and

all a. In our setting of FK percolation for 1 < q < 4 at p = pc, the classical BK

71

inequality does not hold; the goal of this section, namely Proposition 3.2.8 and

Lemmas 3.2.10–3.2.11, is to use an iterative revealing procedure of nested bridges

to nonehthess obtain such a bound for |Be|.

It then follows (see Corollary 3.2.9) that if we sample from the FK measure on

a 2n× 2n box Λ with arbitrary boundary conditions, the boundary conditions this

induces on the concentric inner n × n box will have order log n distinct bridges

over a given edge with probability 1−O(n−c). A simpler formulation of that is as

follows.

Theorem 3.2.2. For every q ∈ [1, 4], there exist K ′ > K > 0 and c(q) > 0 such

that, the critical FK model has, for every e ∈ ∂Λn,n a, say, n10

distance from a

corner of Λn,n,

πZ2,pc,q

(K log n ≤ |Be(ω)| ≤ K ′ log n

)≥ 1−O(n−cK

′) .

The lower bound on |Be| demonstrates that the behavior of the number of

bridges at p = pc is truly different than at p 6= pc; there, by the exponential decay

of correlations at p < pc (dual connections at p > pc), the typical number of bridges

over an edge is O(1).

Relation to mixing estimates. To be more precise about the obstacle posed by

having multiple bridges over an edge, recall the following. In [64] and then [42], the

upper bounds on the mixing time of the Potts models at β = βc for q ∈ 2, 3, 4

relied on RSW bounds [33, 34] to expose dual-interfaces in the FK representation,

beyond which block dynamics chains could be coupled. However, the fact that

chains, started from any two initial configurations, could be coupled past a dual-

interface, relied on a certain conditional event, implicit in the relation between the

72

FK and Potts models at integer q (that no distinct boundary components were

connected in the interior configuration). Without this conditioning, connections

between two components on one side of a rectangle alter the boundary conditions

elsewhere via bridges over the dual-interface, preventing coupling (see Fig. 1.5).

Similar difficulties were pointed out in [9] at p < pc; we also deal with them

in Chapters 7–8 at criticality when q > 4. In both of these cases, the exponential

decay of correlations under π0Z2 ensures that all such bridges would be, with

high probability, microscopic: in [9] these bridges are negligible after restricting

attention to side-homogenous (wired or free on sides) boundary conditions, while in

Chapters 7–8, relevant boundary segments will be disconnected from one another

by brute-force modifications. In contrast, in the present setting at the critical point

of a continuous phase transition, the power-law decay of correlations precludes such

techniques; thus, at p = pc(q) obtaining sharp bounds on the number of bridges

becomes not only necessary, but also substantially more delicate.

3.2.1 Boundary bridges

In this subsection we define boundary bridges of the FK model and related

notation. As explained above, the presence of boundary bridges will be the key

obstacle to coupling and, in turn, to mixing time bounds.

Definition 3.2.3. Consider a rectangle Λ = Λn,n′ with boundary conditions ξ, and

a connected segment L = Ja, bK × n′ ⊂ ∂nΛ. A component γ ⊂ ∂nΛ of ξ is a

bridge over L if there exist v = (v1, v2), w = (w1, w2) ∈ γ such that vξ←→ w and

v1 < a and w1 > b .

73

hulle(γi)hullw(γi)

γi

γi+1

e

Figure 3.3: A pair of boundary bridges, γi, γi+1, over e ∈ ∂nR induced by aconfiguration on Λ−R, and separated by a dual-bridge over e.

Note that every two distinct bridges γ1 6= γ2 over L are disjoint in ξ. Denote by

BL = BL(ξ) the set of all bridges over the segment L. Define bridges on subsets of

∂Λs, ∂Λe, ∂Λw analogously.

Definition 3.2.4 (hull and length of a bridge). The west and east hulls of a bridge

γ over L = Ja, bK× n′ are defined as

hullw(γ) = Jmaxx ≤ a : (x, n′) ∈ γ, aK× n′ ,

hulle(γ) = Jb,minx ≥ b : (x, n′) ∈ γK× n′ ,

so that the hulls of a bridge γ are connected subsets of ∂nΛ (see Fig. 3.3. The west

and east lengths of γ are defined to be

`w(γ) = | hullw(γ)| , `e(γ) = | hulle(γ)| .

Given the above convention, for any L and ξ we can define a east-ordering of

74

BL(ξ) as (γ1, γ2, ..., γ|BL|) where, for all i < j,

`e(γi) < `e(γj) .

Note that, in this ordering of the bridges, hulle(γi) ( hulle(γj) for all i < j. Define

a west-ordering of BL analogously.

Definition 3.2.5. For a subset R ⊂ Λ, an induced boundary condition on ∂R is

one that can be identified with the component structure of an edge configuration

ωΛ−Ro along with the boundary condition on Λ.

Using the above definitions, and planarity, one can check the following useful

facts (depicted in Fig. 3.3). For concreteness we use the east-ordering of BL =

γ1, ..., γ|BL|.

Fact 3.2.6. Let Λ ⊃ R with boundary conditions ξ, and let L ⊂ ∂nR. If γi, for

i < |BL|, is the i-th bridge in the east-ordering of BL, then either the two connected

components of ∂nR− (hullw(γi) ∪ L ∪ hulle(γi)) are connected in Λ−R, or each of

these components is connected to ∂Λ in Λ−R.

Fact 3.2.7. Let Λ ⊃ R with boundary conditions ξ, and let L ⊂ ∂nR. For every

two induced bridges γ1 6= γ2 over a segment L such that hulle(γ1) ⊂ hulle(γ2), either

the two sets (hullw(γ2)4 hullw(γ1)) and (hulle(γ2)4 hulle(γ1)) are dual-connected

in Λ−R, or each of these sets is dual-connected to ∂Λ in Λ−R.

3.2.2 Estimating the number of boundary bridges

In this section, we bound the number of distinct induced boundary bridges over

a segment of ∂R.

75

When sampling boundary conditions on R ⊂ Λ under πξΛ, the induced bridges

over e and all properties of them, are measurable w.r.t. ωΛ−Ro . For any configura-

tion ω, we denote by Be = Be(ωΛ−Ro , ξ) the set of all bridges over e corresponding

to that configuration on Λ, with the above defined west and east orderings.

The main estimate on |Be|, that will be key to the proof of Theorem 9, is the

following.

Proposition 3.2.8. Let q ∈ (1, 4] and fix α ∈ (0, 1]. Consider the critical FK

model on Λ = Λn,n′ with n′ ≥ bαnc, along with the subset R = Λn,n′/2. There exists

c(α, q) > 0 such that for every e ∈ ∂nR, every boundary condition ξ, and every

K > 0,

πξΛ(ω : |Be| ≥ K log n) . n−cK . (3.2.1)

Moreover, there exists c′(α, q) > 0, and for every ε > 0 there is some K0(ε), such

that for every e ∈ Jnε, n − nεK × bn′2c, every boundary condition ξ, and every

K < K0,

πξΛ(ω : |Be| ≥ K log n) & n−c′K .

In addition to the tail behavior of |Be|, we can also classify its typical behavior,

showing that a fixed edge indeed has order log n bridges over it with high probability

(cf. the case of p 6= pc(q) where this quantity is typically O(1)).

Corollary 3.2.9. Let q ∈ (1, 4] and α ∈ (0, 1]. Consider a rectangle Λ = Λn,n′

with n′ ≥ bαnc, along with R = Λn,n′/2. There exists c(α, q) > 0, and for every

ε > 0, there exist K ′ > K(ε) > 0, such that for every e ∈ Jnε, n− nεK×bn′2c and

every ξ,

πξΛ (|Be| /∈ JK log n,K ′ log nK) ≤ n−c .

76

The model at p = pc(q), q ∈ (1, 4] is believed to be scale-invariant; in line with

this, having nested bridges whose length grow exponentially induces c log n clusters

ranging in scale between O(1) and O(nε). Indeed, this is how the lower bound of

Corollary 3.2.9 is obtained. It will, therefore, be important for us to split the set

Be into those bridges according to their proximity to their interior bridges, as well

as the boundary.

For the rest of this subsection, since e is fixed, if e is in the left half of ∂nR then

we will use the east-ordering of Be and otherwise we will use the west-ordering

of Be. If e is in the left half of ∂nR define the subsets Be1 = Be

1(ωΛ−Ro , ξ) and

Be2 = Be

2(ωΛ−Ro , ξ) of Be as follows:

Be1 =

γi ∈ Be : `e(γi−1) ≤ n

6, `e(γi) ≤ 2`e(γi−1)

,

Be2 =

γi ∈ Be : `e(γi−1) ≥ n

6, n− x− `e(γi) ≥ 1

2(n− x− `e(γi−1))

.

For e in the right half of ∂nR, define Be1 and Be

2 analogously, by replacing `e with

`w and n − x with x. For convenience, let γ0 be the possibly nonexistent bridge

given by the two vertices incident to the edge e, which will allow us to treat γ1 as

we would treat the other γi’s.

Before proving Proposition 3.2.8, we present the two lemmas central to the

upper bound (3.2.1) of Proposition 3.2.8, proving exponential tails on each of |Be1|

and |Be2| beyond O(log n). Together, these will imply the O(log n) upper bound

on |Be|, so we defer the proofs of the two lemmas until after completing the proof

of Proposition 3.2.8 using the lemmas. We conclude this section with a proof of

Corollary 3.2.9.

Lemma 3.2.10. There exists c1(α, q) > 0 such that for every e ∈ ∂nR, ξ and

77

K > 0,

πξΛ (ω : |Be1| ≥ K log n) . n−c1K .

Lemma 3.2.11. There exists c2(α, q) > 0 such that for every e ∈ ∂nR, ξ and

K > 0,

πξΛ (ω : |Be2| ≥ K log n) . n−c2K .

With these two lemmas in hand the proof of Proposition 3.2.8 is greatly simpli-

fied.

Proof of Proposition 3.2.8. We begin with the upper bound. Fix an edge

e ∈ ∂nR and a boundary condition ξ on ∂Λ. Without loss of generality suppose

that e is in the left half of ∂nR and use the east-ordering of Be = γ1, γ2, ..., γ|Be|.

Observe that violating the second condition in Be1 means that `e(γi) has at least

doubled the length of its predecessor, whereas violating the second condition in

Be2 means that n − x − `e(γi) is at most half the corresponding quantity of its

predecessor. Noting that violating the length condition of `e(γi−1) (compared to

n/6) is disjoint between Be1 and Be

2, and since `e(γi) ≤ n and n− x− `e(γi) ≥ 1

for all i, we deterministically have

|Be − (Be1 ∪Be

2)| ≤ 2 log2 n ≤ 3 log n .

Using a union bound,

πξΛ (|Be1 ∪Be

2| ≥ (K − 3) log n) ≤ πξΛ(|Be

1| ≥ K−32

log n)

+ πξΛ(|Be

2| ≥ K−32

log n).

The bounds on the two terms on the right-hand side are given by Lemmas 3.2.10–

78

3.2.11, respectively. Taking the minimum of c1, c2 in those lemmas then implies

that there exists c(α, q) > 0 such that

πξΛ(|Be| ≥ K log n) . n−c(K−3)/2 .

In order to prove the lower bound, for any ε > 0, fix any edge e = (x, bn′2c) with

x ∈ Jnε, n− nεK. For i ≥ 1, suppressing the dependence on e, define the sets

Rni = Jx− 2i+1, x+ 2i+1K× Jbn′

2c+ 2i, bn′

2c+ 2i+1K ,

Rei = Jx+ 2i, x+ 2i+1K× Jbn′

2c − 2i, bn′

2c+ 2i+1K , (3.2.2)

Rwi = Jx− 2i+1, x− 2iK× Jbn′

2c − 2i, bn′

2c+ 2i+1K ,

and their respective subsets,

Rni = Jx− 2i+1 + 2i−1, x+ 2i+1 − 2i−1K× Jbn′

2c+ 2i, bn′

2c+ 2i + 2i−1K ,

Rei = Jx+ 2i, x+ 2i + 2i−1K× Jbn′

2c, bn′

2c+ 2i + 2i−1K , (3.2.3)

Rwi = Jx− 2i − 2i−1, x− 2iK× Jbn′

2c, bn′

2c+ 2i + 2i−1K .

When K < K0 := ε log 44

, for every i ≤ 2K log n, we have Rwi , R

ni , R

ei ⊂ Λ. Also

define the following crossing events.

Ai = Cv(Rwi ) ∩ Ch(Rn

i ) ∩ Cv(Rei ) ,

A∗i = C∗v(Rwi ) ∩ C∗h(Rn

i ) ∩ C∗v(Rei ) .

79

Then by definition of distinct bridges in Λ−Ro, we observe that for each k,

|Be| ≥ K log n ⊃K logn⋂i=1

A2i−1 ∩ A∗2i . (3.2.4)

By monotonicity in boundary conditions, the FKG inequality, and Theorem 3.1.2,

there exists p(α, q) > 0 such that for every i ≤ 2K log n,

πξΛ(Ai) ≥ π0Rwi(Cv(Rw

i ))π0Rni(Ch(Rn

i ))π0Rei(Cv(Re

i )) ≥ p ,

πξΛ(A∗i ) ≥ π1Rwi(C∗v(Rw

i ))π1Rni(C∗h(Rn

i ))π1Rei(C∗v(Re

i )) ≥ p .

Thus, if K < K0, we have πξΛ(|Be| ≥ K log n) ≥ p2K logn, as required.

We now prove Lemmas 3.2.10–3.2.11, whose proofs constitute the majority of

the work in obtaining Proposition 3.2.8.

Proof of Lemma 3.2.10. Assume without loss of generality that e = Jx−1, xK×

bn′2c is such that x ≤ n

2and use the east-ordering of Be = γ1, γ2, ..., γ|Be|. In

order to obtain an upper tail on |Be1|, let us describe a revealing procedure for the

FK configuration ω on Λ−Ro under πξΛ.

Let F = J0, nK × Jbn′2c, n′K = Λ − Ro (so that ωF , ξ is the set of connections

with respect to which the existence/properties of bridges are measurable). We can

sequentially reveal γ1, . . . , γ|Be| by exposing the open clusters (in F ) containing

vertices of ∂nΛ one at a time, starting from those adjacent to the right-vertex of e.

Such procedures for exposing the clusters have been used in related settings (see,

e.g., [9, 64]); we formally describe the procedure here since in our case it involves

long-range interactions imposed by the FK boundary conditions. One can reveal

the open cluster Cv containing a vertex v in the set F by

80

1. initializing the set C = v;

2. exposing the values of ω on all edges in E(F ) that contain vertices in C;

3. adding to C any vertices that are now connected by a path of open edges to v

(including possibly the connections imposed by the boundary condition ξ);

4. repeating the process from step (1) with the new set C.

Any open cluster containing vertices in ∂nR on both the right and left sides of e

is a bridge over e. In order to reveal the first m bridges over the edge e, we can

iteratively reveal the open clusters of ∂nR in F , starting initially with the cluster of

(x, bn′2c), and continuing to the right along ∂nR, until m distinct bridges have been

exposed. Using this revealing procedure, the edges which are revealed in order to

expose the first m bridges over e are either enclosed by γm and ∂nR, or belong to

the outer boundary of γm and are closed, thus forming a bounding dual-path.

Let (Fm) be the filtration associated with the above revealing process for the

bridges (Fm reveals γ1, . . . , γm) over the edge e. Our aim is to prove that for every

m ≥ 1,

πξΛ(γm ∈ Be1 | Fm−1) ≤ p , (3.2.5)

(if γm doesn’t exist, we vacuously say γm /∈ Be1) for the choice of

p = 1− p1p2p3 < 1 , (3.2.6)

where p1(α, q), p2(α, q), p3(α, q) > 0 are defined as follows:

• p1 is given by Theorem 3.1.1 with aspect ratio 1 for 1 < q < 4 and by

81

Lemma 3.1.7 with the choice ε = 1/2 and aspect ratio 1/2 for q = 4,

• p2 is the probability given by Theorem 3.1.2 for ε = 1/4 and aspect ratio

6 ∨ α−1,

• p3 is the probability given by Theorem 3.1.2 for ε = 1/3 and aspect ratio 1.

Let us first conclude the proof of Lemma 3.2.10 given (3.2.5). By iteratively

conditioning on (Fi)i≥1 we see that the sequence of indicators (1γi ∈ Be1)i≥1 is

stochastically dominated by the i.i.d. sequence (Zi)i≥1 where Zi ∼ Bernoulli(p). At

the same time, by definition of the set Be1, through this revealing process, as soon

as dlog2 ne many of the indicators (1γi ∈ Be1) are zero, all subsequent ones are

deterministically zero (note that once `e(γi) > n/6, every subsequent bridge will

also have this property). Therefore, we can bound

πξΛ(|Be1| ≥ r) ≤ P

(Bin(r + dlog2 ne − 1, p) ≥ r

)which, upon taking r = K log n and using, say, Hoeffding’s inequality once K ≥

2p−1, yields the desired estimate.

We now turn to proving the conditional estimate of (3.2.5). First observe that

by Fact 3.2.7 and the definition of hulle(γm−1), if Lw, Le are the two connected

subsets of ∂nR− hulle(γm−1), the event

Em =Lw

F ∗←→ Le or LeF ∗←→ ∂Λ

satisfies Em ⊃ |Be| ≥ m ⊃ γm ∈ Be1. In fact, the revealing process of Fm−1

reveals precisely the dual-path that bounds the open cluster of γm−1, and that

dual-path is either a dual connection from Lw to Le in F ∗, or it is the west-most

82

n′2

n′2

+ k

n′

x x + k x + 2k x + 3kz

Λ−R

ζ

R3

R2

γm−1

R1

Figure 3.4: After conditioning on ζ (via the configuration in the blue shadedregion), the probability of the purple and green dual-crossings is greater thanp1p2p3, bounding the probability of γm ∈ Be

1.

dual crossing from Le to ∂Λ that is to the right of γm−1. Either way, denote by ζ

the dual-bridge/crossing revealed as such by Fm−1 (see Fig. 3.4), and let (z, bn′2c)

be the west-most point of ζ ∩ ∂nR.

Let k = `e(γm−1); in order for γm ∈ Be1, necessarily `e(γm) ≤ 2k and

(z, bn′2c) ∈ Jx+ k, x+ 2kK× bn′

2c =: I .

We will establish the desired upper bound of (3.2.5) uniformly over Fm−1, ζ and

k. It suffices to only consider k ≤ n6

because otherwise, `e(γm) > n6

and therefore

γm /∈ Be1 deterministically.

Note that conditional on Fm−1 (which contains the σ-algebras generated by ζ

and k), by Fact 3.2.6, the event γm ∈ Be1 implies the event S, stating that either

ζ is a dual-bridge and I is primal-connected in F ∪ ξ to the left component of

∂nR−hulle(ζ), or alternatively ζ is a dual-crossing to ∂Λ and I is primal-connected

83

to ∂Λ in F . Thus, in this conditional space,

γm /∈ Be1 ⊃

ζ

F ∗←→ Jx+ 2k, x+ 3kK× bn′2c, (3.2.7)

since the right-hand side of Eq. (3.2.7) implies Sc which implies the left-hand side.

In order to lower bound the probability of the last event in Eq. (3.2.7), let D∗

be the outer (if ζ is a dual-crossing in F , then eastern) connected component of

F ∗ − ζ, and let D be its dual. Define also the following subsets of Λ:

R1 =Jx, x+ kK× Jbn′2c, bn′

2c+ mink, αn

4K ,

R2 =Jx, x+ 3kK× Jbn′2c+ mink

2, αn

8, bn′

2c+ mink, αn

4K ,

R3 =Jx+ 2k, x+ 3kK× Jbn′2c, bn′

2c+ mink, αn

4K ,

whereby, the event in the right-hand side of Eq. (3.2.7) can be written as ζ F ∗←→

∂sR3. For any i = 1, 2, 3, define the following crossing events (see Fig. 3.4):

C∗v(Ri ∩D) =∂eRi

Ri∩D←→6 ∂wRi

,

C∗h(Ri ∩D) =∂nRi

Ri∩D←→6 ∂sRi

.

(3.2.8)

(observe that implicit in (C∗v(Ri∩D))c is the event ∂eRi∩D 6= ∅∩∂wRi∩D 6= ∅,

and similarly, implicit in (C∗h(Ri∩D))c is the event ∂nRi∩D 6= ∅∩∂sRi∩D 6= ∅).

Claim 3.2.12. Conditional on Fm−1 (and in particular also ζ and k),

γm /∈ Be1 ⊃

(C∗v(R1 ∩D) ∩ C∗h(R2 ∩D) ∩ C∗v(R3 ∩D)

).

Proof. Suppose that ω satisfies the events on the right-hand. Recall that ζ is

84

such that ∂eR3 ∩ D 6= ∅ and ∂wR3 ∩ D 6= ∅, and ∂eR3←→6 ∂wR3 in R3 ∩ D since

ω ∈ C∗v(R3 ∩D). Consider R3 ∩D with boundary conditions wired on ∂e,wR3 ∩D

and free on ζ and ∂n,sR3 ∩D; then the boundary conditions on R3 ∩D alternate

between free and wired on boundary curves ordered clockwise as Lw1 , Lf1 , L

w2 , L

f2 ...;

by planarity and the choice of generalized Dobrushin boundary conditions, for

any two wired boundary curves Lwi , Lwi+1, either Lwi ←→ Lwi+1, or Lfi

∗←→ Lfj for

some j 6= i. Picking the two wired segments of ∂e,wR3 ∩ D closest to ∂sR3, the

aforementioned fact that ∂eR3←→6 ∂wR3 in R3 ∩D implies that either ∂sR3∗←→ ζ

or ∂sR3∗←→ ∂nR3. In the former, γm /∈ Be

1 holds by Eq. (3.2.7), so suppose only

the latter holds and call the dual-crossing ζ3.

Since ∂sR3∗←→ ∂nR3, both ∂sR2 ∩D and ∂nR2 ∩D are nonempty. Clearly, ζ3

splits R2 ∩D into the subset to its east, Ue, and that to its west, Uw. Consider

the set to its east, Ue, with boundary conditions that are wired on ∂s,nR2 ∩D and

free on ζ and on ∂e,wR2 ∩D. Since ζ3 and ζ are vertex-disjoint (by our assumption

that ∂sR3∗←→6 ζ in R3 ∩D), and the wired boundary segments adjacent to ζ3 are

disconnected in Ue, it must be that either ζ3∗←→ ζ or ζ3

∗←→ ∂eR2 in Ue. Using

the same reasoning on Uw, either ζ3∗←→ ζ or ζ3

∗←→ ∂wR2 in Uw. Combining these,

either ζ∗←→ ζ3, in which case ζ

∗←→ ∂sR3, or alternatively ∂eR2∗←→ ζ3

∗←→ ∂wR2

in R2 ∩D. In the former case, by Eq. (3.2.7), γm /∈ Be1; assume therefore that

only the latter case holds, and let ζ2 be a dual-crossing between ∂eR2 to ∂wR2 that

intersects ζ3.

Finally, we can deduce that ∂eR1 ∩D and ∂wR1 ∩D are nonempty as ζ2 and ζ

are vertex-disjoint (by our assumptions ζ∗←→6 ζ3 and ζ2

∗←→ ζ3). Considering now

Us, the subset of R1 ∩D south of ζ2 with wired boundary conditions on ∂e,wR1 ∩D

and free elsewhere, as before we deduce that either ζ2∗←→ ζ or ζ2

∗←→ ∂sR1 in Us.

85

Since, by definition of ζ, deterministically ∂sR1 ∩D = ∅, the former must hold, and

ζ∗←→ ∂sR3 through ζ2 and ζ3, and Eq. (3.2.7) concludes the proof.

We will next bound the probability of each of the events C∗v(R1∩D), C∗h(R2∩D)

and C∗v(R3 ∩ D), which, using the above claim, will translate to a bound on

γm /∈ Be1.

To see this, first note that by planarity, for all i = 1, 2, 3 and every subset D,

C∗v(Ri ∩D) ⊃ (Ch(Ri))c = C∗v(Ri) , (3.2.9)

and likewise for horizontal crossing events. Define the rectangle R1 ⊃ R1 by

R1 = Jx, x+ kK× Jbn′2c, n′K ⊂ Λ .

Let the boundary conditions (1, 0) on R1 be free on ∂sR1 and wired on ∂n,e,wR1.

Combining Eq. (3.2.9), monotonicity in boundary conditions, and the domain

Markov property, we get for p1(α, q) > 0 given by Eq. (3.2.6),

πξΛ(C∗v(R1 ∩D) | `e(γm−1) = k,Fm−1, ζ

)≥ π1

R1(C∗v(R1 ∩D) | `e(γm−1) = k,Fm−1, ζ)

≥ π1,0

R1(C∗v(R1)) ≥ p1 ,

where the last inequality follows from Theorem 3.1.1, Lemma 3.1.7 and self-duality.

We stress that wiring of ∂n,e,wR1 allowed us to ignore the information revealed on

R1 −D as far as the configuration in R1 ∩D is concerned, and the fact that ωζ is

closed allowed us to place a free boundary on ∂sR1, supporting Lemma 3.1.7.

86

Next, consider the rectangle R2 ⊃ R2 defined by

R2 = Jx− k, x+ 4kK× Jbn′2c, n′K ,

so that R2 ⊂ Λ since k = `e(γm−1) ≤ n/6. By monotonicity in boundary conditions

and Eq. (3.2.9), we get that for the choice of p2(α, q) > 0 given by Eq. (3.2.6),

πξΛ(C∗h(R2 ∩D) | `e(γm−1) = k,Fm−1, ζ)

≥ π1R2

(C∗h(R2 ∩D) | `e(γm−1) = k,Fm−1, ζ)

≥ π1R2

(C∗h(R2)) ≥ p2 .

Similarly, applying the exact same treatment of R2 to

R3 = Jx+ k, x+ 4kK× Jn′4, n′K ⊂ Λ ,

(it is possible to encapsulate R3 by a rectangle with wired boundary conditions

since ζ does not intersect ∂sR3 in our conditional space) shows that

πξΛ(C∗v(R3 ∩D) | `e(γm−1) = k,Fm−1, ζ)

≥ π1R3

(C∗v(R3 ∩D) | `e(γm−1) = k,Fm−1, ζ)

≥ π1R3

(C∗v(R3)) ≥ p3 ,

for p3(α, q) > 0 as defined in Eq. (3.2.6).

Putting these all together, by the FKG inequality and Claim 3.2.12,

πξΛ(γm /∈ Be

1

∣∣ Fm−1

)≥ p1p2p3 ,

87

n′2

n′2

+ l4

n′

n− 7l6

n− l n− l3

n− l6

ζ

R3

R2

γm−1

R1

Figure 3.5: After revealing ζ (via the blue shaded region), the existence of the threedual-crossings depicted precludes |Be

2| ≥ m.

implying the desired (3.2.5), and concluding the proof.

Proof of Lemma 3.2.11. Without loss of generality suppose e is in the left half

of ∂nR and use the east-ordering of bridges so that Be = γ1, ..., γ|Be|.

The proof follows the same argument used to prove Lemma 3.2.10. In what

follows we describe the necessary modifications that are needed here. Recall the

prescribed revealing process for the configuration on F = J0, nK× bn′2c described in

the proof of Lemma 3.2.10; recall also that (Fm) is the filtration corresponding to

the process of sequentially revealing the distinct bridges over the edge e. Our goal

is to prove the following analogue of (3.2.5), that for every m ≥ 1,

πξΛ(γm ∈ Be2 | Fm−1) ≤ p , (3.2.10)

for the choice of p = 1− p1p2p3 < 1 where,

• p1 is given by Theorem 3.1.1 with aspect ratio α for 1 < q < 4 and by

Lemma 3.1.7 with the choice ε = 1/2 and aspect ratio α/2 for q = 4 ,

88

• p2 is the probability given by Theorem 3.1.2 for ε = 1/8 and aspect ratio 6/α ,

• p3 is the probability given by Theorem 3.1.2 for ε = 1/3 and aspect ratio α .

Indeed, by iteratively conditioning on (Fi)i≥1, the bound (3.2.10) allows us to

stochastically dominate the sequence of indicators (1γi ∈ Be1)i≥1 by the i.i.d.

sequence (Zi)i≥1 where Zi ∼ Bernoulli(p), and moreover by definition of Be2, as soon

as dlog2 ne of the indicators are zero, all subsequent ones are deterministically zero.

The desired inequality then follows by comparison to P(Bin(r+ dlog2 ne− 1, p) ≥ r)

for r = K log n.

As before, we consider a fixed m, and let Lw, Le be the left and right connected

components of ∂nR− hulle(γm−1). As in the proof of Lemma 3.2.10, reveal γm−1,

in which case we reveal the enclosing dual-path ζ attaining

Em =Le

F ∗←→ Lw or LeF ∗←→ ∂Λ

,

whose west-most vertex of intersection with ∂nR is marked by (z, bn′2c). Condition-

ally on Fm−1, which contains the σ-algebras of γm−1, ζ and `e(γm−1) = k,

Also for any instance of the configuration revealed by Fm−1, we can set k =

`e(γm−1) as before, and let

l := n− (x+ `e(γm−1)) .

If n−z < l/2, deterministically γm /∈ Be2 (as argued in the proof of Proposition 3.2.8),

hence we may assume that n − z ≥ l/2; moreover, since k ≥ n6, it must be that

89

l6≤ k. Define the following subsets of Λ:

R1 =Jn− l − l6, n− lK× Jbn′

2c, bn′

2c+ αl

6K ,

R2 =Jn− l − l6, n− l

6K× Jbn′

2c, bn′

2c+ αl

6K ,

R3 =Jn− l3, n− l

6K× Jbn′

2c, bn′

2c+ αl

6K .

Define C∗v(R1 ∩D), C∗h(R2 ∩D), C∗v(R3 ∩D) as in Eq. (3.2.8). As in Claim 3.2.12,

γm /∈ Be2 ⊃

(C∗v(R1 ∩D) ∩ C∗h(R2 ∩D) ∩ C∗v(R3 ∩D)

).

Finally, for Ri, i = 1, 2, 3 given by

R1 =Jn− l − l6, n− lK× Jbn′

2c, bn′

2c+ αl

3K ,

R2 =Jn− l − l3, nK× Jbn′

2c − αl

6, bn′

2c+ αl

3K ,

R3 =Jn− l2, n, l

2K× Jbn′

2c − αl

6, bn′

2c+ αl

3K ,

(note that all three are subsets of Λ, by the fact that l ≤ n and n′ ≥ bαnc), the

same monotonicity argument used in the proof of Lemma 3.2.10 now implies (see

Fig. 3.5) that

πξΛ(γm−1 /∈ Be

2

∣∣ Fm−1

)≥ p1p2p3 ,

implying (3.2.10) and concluding the proof.

By matching the tail estimate of Proposition 3.2.8 with a lower bound, we can

straightforwardly see that an order log n bridges over a fixed edge is indeed typical.

Proof of Corollary 3.2.9. Fix an abitrary ε > 0, any boundary condition ξ, and

90

any e ∈ Jnε, n − nεK × bn′2c. For this e, recall the definitions of the rectangles

Rni , R

ei , R

wi as well as their subsets Rn

i , Rei , R

wi from (3.2.2)–(3.2.3). As before, when

M < M0 := ε log 44

, for every i ≤ 2M log n, all these are subsets of Λ and we can

define the crossing events

Ai = Cv(Rwi ) ∩ Ch(Rn

i ) ∩ Cv(Rei ) , and A∗i = C∗v(Rw

i ) ∩ C∗h(Rni ) ∩ C∗v(Re

i ) .

Now for each i ≤ M log n, we can define the event χi := A2i−1 ∩ A∗2i and notice

that

|Be| ≥M logn∑i=1

1χi .

Observe that for each i, the event Ai is measurable with respect to the configuration

ω on the half-annulus Rwi ∪Rn

i ∪Rei . By a similar reasoning as before, there exists

some p = p(α, q) > 0 such that for every i = 1, ..., 2M log n, and every configuration

η,

πξΛ(Ai | ωΛ−Rn,e,wi

= η) ≥ π0Rwi(Cv(Rw

i ))π0Rei(Cv(Re

i ))π0Rni(Ch(Rn

i )) ≥ p ,

πξΛ(A∗i | ωΛ−Rn,e,wi

= η) ≥ π1Rwi(C∗v(Rw

i ))π1Rei(C∗v(Re

i ))π1Rni(C∗h(Rn

i )) ≥ p .

Observe that for i 6= j the interiors of Rn,e,wi and Rn,e,w

j are disjoint. As a

consequence, we can also deduce by the domain Markov property and monotonicity,

that for every configuration η, for every i = 1, ...,M log n,

πξΛ(χi | ωΛ−Rn,e,w

2i−1−Rn,e,w2i

= η)≥ p2 .

91

In particular, the sequence of indicators (1χi)i=1,...,M logn stochastically dominates

a sequence of i.i.d. Bernoulli(p2) random variables. We therefore deduce that

|Be| ≥M logn∑i=1

1χi Bin(M log n, p2)

Choosing K < p2

2M , and using Hoeffding’s inequality to bound the probability that

the binomial random variable on the right-hand side is at most K log n, we see that

πξΛ(|Be| ≤ K log n

)≤ exp

[− 1

4Mp4 log n

].

Combining this via a union bound with the upper bound from (3.2.1) in Proposi-

tion 3.2.8 implies the desired.

3.2.3 Disjoint crossings

To extend our FK mixing time bound from favorable boundary conditions

(see §4.2.4) to periodic boundary conditions (which are not in that class) in §4.2.6,

we will need an analogous bound on the number of disjoint crossings of a rectangle.

For a rectangle R = (V (R), E(R)) and a configuration ωE(R), let ΨR =

ΨR(ωE(R)) be the set containing every component A ⊂ V (R) (connected via the

edges of ωE(R)) that intersects both ∂sR and ∂nR. We will need the following

equilibrium estimate similar to Proposition 3.2.8.

Proposition 3.2.13. Let q ∈ (1, 4] and α ∈ (0, 1]. Consider the critical FK model

on Λ = Λn,n′ with n′ ≥ bαnc, and the subset R = J0, nK × Jn′3, 2n′

3K. There exists

92

c(α, q) > 0 such that for every boundary condition ξ and every m ≥ 3,

πξΛ (|ΨR| ≥ m) ≤ e−cm .

Proof. We will prove by induction that, for all m ≥ 1,

πξΛ(|ΨR| ≥ m) ≤ (1− p)m−2 , (3.2.11)

where p > 0 is as given by Theorem 3.1.1 with aspect ratio 3/α when 1 < q < 4,

and is as given by Corollary 3.1.8 with aspect ratio α/3 when q = 4.

The cases m = 1, 2 are trivially satisfied for any 0 < p < 1. Now let m ≥ 3,

and suppose that Eq. (3.2.11) holds for m− 1; the proof will be concluded once we

show that

πξΛ(|ΨR| ≥ m

∣∣ |ΨR| ≥ m− 1)≤ 1− p .

Conditioned on the existence of at least m− 1 distinct components in ΨR, we can

condition on the west-most component in ΨR (by revealing all dual-components

of ωR incident to ∂wR, then revealing the primal-component of the adjacent

primal-crossing). We can also condition on the m − 2 east-most components in

ΨR (by successively repeating the aforementioned procedure from east to west,

i.e., replacing ∂wR above by ∂eR to reveal some component C ∈ ΨR, then by its

western boundary ∂wC, etc.).

Through this process, we can find two disjoint vertical dual-crossings ζ1, ζ2 of

R, each one a simple dual-path; the set (R∗ − ζ1 − ζ2)∗ consists of three connected

subsets of R; let D denote the middle one. There are exactly m− 1 elements of

ΨR in R−D, thus its m-th element, if one exists, must belong to D. Since every

93

edge in ζ1 ∪ ζ2 is dual-open, for any such choice of ζ1, ζ2, we then have

πξΛ(|ΨR| ≥ m

∣∣ |ΨR| ≥ m− 1, ζ1, ζ2

)= πξΛ

(Cv(D)

∣∣ ζ1, ζ2

),

Using the domain Markov property and monotonicity of boundary conditions,

πξΛ(Cv(D)

∣∣ ζ1, ζ2

)≤ π1,0,1,0

D (Cv(D)) ,

where (1, 0, 1, 0) boundary conditions on D denote those that are free on ζ1, ζ2 and

wired on ∂R ∩D. Again by monotonicity (in boundary conditions and crossing

events),

π1,0,1,0D (Cv(D)) ≤ π1,0,1,0

R

(Cv(D)

∣∣ ωζ1 = 0, ωζ2 = 0)≤ 1− π1,0,1,0

R (C∗h(R)) ,

where, following the notation of Corollary 3.1.8, (1, 0, 1, 0) boundary conditions on

a rectangle R are wired on ∂n,sR and free on ∂e,wR. By monotonicity in boundary

conditions and the definition of p, the right-hand side is bounded above by

1− π(1,0,1,0)R

(C∗h(J0, nK× Jn′

3, n′

3+ αn

3K))≤ 1− p .

94

Chapter 4

Mixing on (Z/nZ)2 at continuous

phase transitions

In this chapter, we analyze the Glauber dynamics of the Potts and FK models

at their critical point when q ∈ (1, 4], so that their phase transition is continuous.

The rich equilibrium picture described at the beginning of Chapter 3 is believed

to present itself in the local-update dynamics of the system by e.g., inducing an

auto-correlation length and spectral gap that are polynomial in the system size,

with universal, dynamical critical exponents (see e.g., [39, 61, 82, 89]).

In the case of q = 2, Holley [50] showed that the continuous-time Ising Glauber

dynamics has an auto-correlation that decays as t−1/4 and deduced that the spec-

tral gap decays at least polynomially in n. Lubetzky and Sly [64] proved that

the inverse spectral gap of the critical Ising Glauber dynamics on Λn,n is truly

polynomial in n (independently of its boundary conditions). They relied on a

recursive block dynamics approach, along with the RSW estimate of [33] to couple

Ising configurations beyond dual-crossings with a positive probability. This implied

95

that the inverse gap on an n× n box is at most a constant multiple of the inverse

gap on an n2× n

2box; recursing O(log n) times yields a polynomial upper bound on

the inverse gap.

In Section 4.1, we use a similar approach to analyze the q = 3, 4 Potts Glauber

dynamics. With the RSW estimates described in Section 3.1 in hand, the argument

of [64] extends to q = 3; at q = 4 we use the lower-bounds on crossing probabilities

from Section 3.1.1 in place of the RSW estimates up to the boundary. Also, though

at q = 3, 4 we do not have access to precise arm exponents as was necessary for the

mixing time lower bounds of [50, 64], we use the lower bounds on the arm exponent

from Section 3.1.2 to give a polynomial lower bound for the inverse spectral gap.

In Section 4.2, we prove corresponding upper bounds on the mixing times of

the FK Glauber dynamics. We would like to adapt a similar strategy, following a

recursive approach and using the RSW estimates at q ∈ (1, 4] to couple beyond a

dual-crossing, whose existence has a positive probability. However, as indicated

in Section 3.2 significant complications arise due to the non-locality of the FK

boundary conditions induced on sub-blocks. These difficulties are discussed in

detail, together with a road map of how we deal with them, in Section 4.2.1.

4.1 Potts Glauber dynamics at continuous phase

transition points

This section contains the proof of Theorem 1 (as well as its analogs for boxes

with non-periodic boundary conditions); recall from Theorem 1.2.2 that it suffices

to prove the desired bounds for Glauber dynamics for the Potts model in order

to obtain them for FK Glauber as well as Swendsen–Wang and Chayes–Machta

96

dynamics at integer q. Consider Λ = Λn,n′ = J0, nK× J0, n′K for n′ = bαnc, where

α ∈ [α, 1] for some fixed 0 < α ≤ 12.

4.1.1 Mixing under arbitrary boundary conditions

We first establish analogues of Eqs. (1.1.1)–(1.1.2) for Glauber dynamics for the

Potts model with arbitrary boundary conditions, modulo an equilibrium estimate

on crossing probabilities at q = 4 which we established in §3.1.1. Whenever we refer

to arbitrary or fixed boundary conditions we mean ones that include an assignment

of a color, or free to each of the vertices of ∂Λ (in contrast to periodic). The

following is a general form of the approach of [64] to proving upper bounds on

mixing times in the presence of RSW bounds; we stress that, while this proof

does extend from the Ising model to the Potts model, in fact it fails to produce a

polynomial upper bound for the critical FK model at noninteger 1 < q < 4, despite

the availability of the necessary (uniform) RSW estimates (cf. Section 4.2).

Theorem 4.1.1. Suppose q ≥ 1 and that there exists a nonincreasing sequence

(an) such that

π0Λn/3,n

(Cv(Λn/3,n)

)≥ an . (4.1.1)

Then there exists some absolute constant c > 0 such that Glauber dynamics for the

Potts model on Λ = Λn,n′ with arbitrary boundary conditions, ζ, satisfies

gap−1p ≤ (c an)−2 log3/2 n .

Combining the RSW bound of Theorem 3.1.1 with Theorem 4.1.1 establishes

97

the analog of Eq. (1.1.1) for a rectangle with arbitrary (non-periodic) boundary con-

ditions. At q = 4, we use the polynomially decaying bound on crossing probabilities

uniform in boundary conditions (see Theorem 3.1.5), through which Theorem 4.1.1

will yield the matching quasi-polynomial upper bound on mixing.

Corollary 4.1.2. There exist absolute constants c1, c2 > 0 such that Glauber

dynamics for the critical 3-state Potts model on Λ with arbitrary fixed boundary

satisfies

gap−1p . nc1 ,

whereas for the 4-state critical Potts model on Λ with arbitrary boundary conditions,

gap−1p . nc2 logn .

From this corollary, Eqs. (1.1.1)–(1.1.2) of Theorem 1 follow by moving from

the box to the torus exactly as done in [64, Theorem 4.4] (see also §4.1.2). For

q = 2, 3, 4, by Theorem 1.2.2, these imply the analogous upper bounds on the

inverse gap of the Swendsen–Wang dynamics, as well as Glauber dynamics for the

FK model.

Proof of Theorem 4.1.1. We use the block dynamics technique of Theorem 2.2.8

used in [64]. Define two sub-blocks of Λ, as follows:

Bw := J0, 2n3

K× J0, n′K , Be := Jn3, nK× J0, n′K .

Then let B denote the block dynamics on Λ with sub-blocks Bw, Be as defined

in §2.2.2. We bound gapB(Λζ) and gapp(Bϕi ) of Theorem 2.2.8 uniformly in ζ, ϕ.

Lemma 4.1.3. For any two initial configurations σ, σ′ on Λ with corresponding

98

block dynamics chains Xt and Yt, there exists an absolute constant c > 0 such that,

if (an) is a sequence satisfying (4.1.1), there is a grand coupling, such that P(X1 6=

Y1) ≤ 1 − c an. Moreover, there exists some c′ > 0 such that gapB(Λζ) ≥ c′ an

uniformly in ζ.

Proof. We construct explicitly a grand coupling that allows us to couple the two

configurations with the above probability. First recall that the Potts boundary

condition ζ on Λ corresponds to an FK boundary condition ξ where two boundary

vertices are in the same cluster if and only if they have the same color, along with

the decreasing event Fζ = ω : ∀x, y ∈ ∂Λ, ζ(x) 6= ζ(y) =⇒ xΛ←→6 y. Via the

Edwards–Sokal coupling, we move from the Potts model with boundary ζ to the

corresponding FK model with boundary ξ conditional on the event Fζ .

Suppose the clock at block Bw rings first. The two initial configurations σ, σ′

induce two Potts boundaries η, η′ corresponding to FK boundaries ψ, ψ′ on ∂eBw

along with the events Fη,ζ and Fη′,ζ ; (η, ζ) is the boundary condition on B1 with η

on ∂eB1 and ζ∂Bwon the rest of ∂Bw. Here and throughout the rest of the paper,

when discussing boundary conditions, we use the restriction to a line to denote the

boundary condition induced on that line by the configuration we have revealed.

We seek to couple the two initial configurations on all of Λ by first coupling

them on Λ − Boe. For each initial configuration, the block dynamics samples a

Potts configuration on Bw by sampling an FK configuration from πψ,ξBw(· | Fη,ζ),

πψ′,ξ

Bw(· | Fη′,ζ).

Via the grand coupling defined in §2.2.1 of all boundary conditions on ∂eBw,

we reveal the open component of ∂eBw in order to condition on the right-most dual

vertical crossing of Λ−Boe. Note that all FK measures we consider are stochastically

dominated by π1,ξBw

(by monotonicity in boundary conditions and since Fη,ζ is a

99

decreasing event). Then if a sample from π1,ξBw

has a dual vertical crossing in Bw∩Be,

under the grand coupling, so will all the samples of πψ,ξBw(· | Fη,ζ).

Under the event Fη,ζ , by construction it is impossible to add boundary connec-

tions by modifying the interior of Bw (either such connections would be between

monochromatic sites in which case they are already in the same cluster, or otherwise

such connections are impossible under Fη,ζ). Thus, if there is such a dual vertical

crossing under π1,ξBw

, the event Fη,ζ ensures that to the west of that crossing, all

realizations of πψ,ξBwsee the same boundary conditions. By the domain Markov

property, the grand coupling then couples all such realizations west of the right-

most dual-crossing of π1,ξBw

and therefore on all of Λ−Boe (for the explicit revealing

procedure, see [64, §3.2]). We then use the same randomness to color coupled

clusters the same way, and couple all corresponding Potts configurations on Λ−Boe.

The colorings of the boundary clusters are predetermined, but because the ∂eBw

boundary clusters cannot extend past the dual vertical crossing, the two Potts

configurations can be coupled west of the dual vertical crossing.

Suppose the clock at block Be rings next. If we have successfully coupled Xt

and Yt in Λ−Boe, then the identity coupling couples the configurations on all of Λ.

But note that by the assumption of Theorem 4.1.1, and the fact that n′ ≤ n,

π1Bw

(C∗v(Jn3 ,

2n3

K× J0, n′K))≥ an .

Moreover, by time t = 1, there is a probability c > 0 that the dynamics rang

the clock of Bw and then the clock of Be in which case we have coupled the two

configurations with probability an at time t = 1. By the submultiplicativity of d(t),

100

for all t > 0,

P(Xt 6= Yt) ≤ (1− c an)t ,

which implies that there exists a new constant c > 0 such that tmix ≤ c/an and

(log 12ε

)(gapB(Λζ))−1 ≤ tmix(ε) ≤ c/an .

In particular there exists c′ > 0 such that (gapB(Λζ))−1 ≤ c′/an.

By Theorem 2.2.8 there exists c > 0 such that we get the following relation

between the gap of Glauber dynamics on Λ and Glauber dynamics on the blocks

Bi, i ∈ e,w:

(gapp(Λζ))−1 ≤(c an)−1 max

imaxσ

(gapp(Bσi )−1.

However, each Bi is a rectangle Λ2n/3,n′ with some fixed boundary conditions and

one can check that for α ∈ [α, 1], it also has, up to rotation, aspect ratio αi ∈ [α, 1].

It follows that maxi maxσ(gapp(Biσ))−1 satisfies the same relation as (gapp(Λ

ζ))−1.

Recursing 2 log3/2 n times yields the desired bound on (gapp(Λζ))−1 for any ζ.

4.1.2 Periodic boundary conditions

We now complete the proof of Theorem 1 by transferring the bounds on mixing

times under fixed boundary conditions to the torus. The translation of the mixing

time bounds is identical to the proof of [64], and we therefore do not lay out the

details. A similar (but more complicated) translation of mixing time bounds from

fixed boundary conditions to the torus, is used in Section 4.2.6, whose full details

are included.

101

Proof of Theorem 1. The proof to go from arbitrary boundary conditions to the

torus is the same as the proof of Theorem 4.4 of [64] which used block dynamics

twice to first reduce mixing on the torus (Z/nZ)2 to a cylinder (Z/nZ)× J0, n′K,

and then that cylinder to a rectangle with fixed boundary conditions, on which

Corollary 4.1.2 gives the desired polynomial (quasi-polynomial) mixing time bound.

We only observe that the proof goes through after replacing the RSW bounds there

by the estimate in Theorem 3.1.5, and conditioning on the event Fζ as before.

4.1.3 Polynomial lower bounds

In order to provide as complete a picture as possible, we also extend the

polynomial lower bound of [64] to the Glauber dynamics for the q = 3, 4 Potts

models, showing that indeed they undergo a critical slowdown. We do not have

access to precise arm exponents as exist for q = 2, so instead we appeal to the

arm-exponent lower bound derived in Lemma 3.1.9.

Theorem 4.1.4. Let q = 3, 4 and consider the critical Potts model on Λ = J0, nK2

with boundary conditions η. The continuous-time Glauber dynamics has gap−1 & cn

for some c(q) > 0, and the same holds on (Z/nZ)2.

Proof. Now that we have a bound on connection probabilities macroscopically

away from boundaries, we modify the lower bound of [64] to our setting. Fix any

boundary condition η (if we are considering the torus, reveal σ∂Λ and fix that to

be your boundary condition η). Let Λ1 = Jn4, 3n

4K2 and Λ2 = Jn

3, 2n

3K2. Recall the

variational form of the spectral gap, Eq. (2.2.2), and consider the test function

f(σ) =∑

x∈Λ21σ(x) = q. Since f is 1-Lipschitz, E(f, f) is easily seen to be

O(n2). We now lower bound Varµ(f). To do so, we move to the FK representation

102

of the Potts model on Λ via the Edwards–Sokal coupling using the event Fη.

For the remainder of this proof only, let E and Var be with respect to the joint

distribution over FK and Potts configurations given by the Edwards–Sokal coupling.

By Theorem 3.1.3, we see by the FKG inequality,

πξΛ(C∗o(Λ− Λ1) | Fη) ≥ π1Λ(C∗o(Λ− Λ1)) ≥ c1

for some c1(q) > 0. By the law of total variance and the above, we see that

Varµ(f) ≥ E[Var(f | ωΛ−Λ1

)1C∗o(Λ− Λ1)]≥ c1Var(f | ωΛ−Λ1

, ∂Λ1←→6 ∂Λ) .

But given that ∂Λ1←→6 ∂Λ, the probability of σ(x) = q for x ∈ Λ2 is 1/q; in

particular, by the Edwards–Sokal coupling and FKG, we can expand the above as

Var(f | ωΛ−Λ1, ∂Λ1←→6 ∂Λ) ≥ q−1

∑x,y∈Λ2

πξΛ(x←→ y | ωΛ−Λ1, ∂Λ1←→6 ∂Λ)

≥ q−1∑x,y∈Λ2

π0Λ1

(x←→ y) ≥ c2n3

for some c2(q) > 0, where the last inequality follows from (3.1.5) of Lemma 3.1.9.

4.2 FK Glauber dynamics at continuous phase

transition points

In this section, we prove Theorem 3. While we would still like to use a recursive

approach based on coupling beyond RSW-type dual-crossings. However, we run

into significant difficulties due to non-localities inherent to the FK model, as can

103

be encoded into what we referred to as boundary bridges in Setion 3.2. To be more

precise about the obstacle posed by having multiple bridges over an edge, recall

the following. In Section 4.1, the upper bounds on the mixing time of the Potts

models at β = βc for q ∈ 2, 3, 4 relied on RSW bounds [33, 34] to expose dual-

interfaces in the FK representation, beyond which block dynamics chains could be

coupled. However, the fact that chains, started from any two initial configurations,

could be coupled past a dual-interface, relied crucially on the conditional event Fζ ,

which was inherent to the Edwards–Sokal coupling of the models with boundary

conditions (that no distinct boundary components were connected in the interior

configuration). Without this conditioning, connections between two components

on one side of a rectangle alter the boundary conditions elsewhere via bridges over

the dual-interface, preventing coupling (see Fig. 1.5).

Moreover, as we saw in Section 3.2, such boundary bridges cannot be avoided—

indeed, as Corollary 3.2.9 indicated, the typical boundary condition that will be

induced on a sub-block in the block dynamics will have many long-range boundary

bridges over any fixed edge.

In Section 4.2.1, we sketch out the proof approach and the necessary modi-

fications to the block-dynamics scheme of the preceding section to allow us to

disregard unfavorable boundary conditions with atypically many bridges. We then

introduce these dynamical tools and their applications, and prove the main theorem

in Sections 4.2.4–4.2.6.

4.2.1 Main techniques

To convert our upper bounds on bridges from Section 3.2 to an upper bound

on the mixing time, our dynamical analysis restricts its attention to boundary

104

conditions with |Be| = O(log n) for every e ∈ ∂Λ; we call such boundary conditions

“typical” (see Definitions 4.2.11–4.2.12) and observe that wired and free boundary

conditions are both typical.

To maintain “typical” (as opposed to worst-case) boundary conditions through-

out the multi-scale analysis, we turn to the Peres–Winkler censoring inequalities [76]

for monotone spin systems, that were used in [70] (then later in [63]) for the Ising

model under “plus” boundary, a class of boundary conditions that dominate the

plus phase (observe that, in contrast, “typicality” is not monotone).

A major issue when attempting to carry out this approach—adapting the

analysis of the low temperature Ising model to the critical FK model—is the stark

difference between the nature of the corresponding equilibrium estimates needed

to drive the multi-scale analysis. In the former, crucial to maintaining “plus”

boundary conditions throughout the induction of [70] was that in the presence of

favorable boundary conditions, the multiscale analysis could be controlled except

with super-polynomially small probability. This yielded a bound on coupling the

dynamics started at the extremal (plus and minus) initial configurations, which a

standard union bound over the O(n2) sites of the box (see (2.2.7)) then transformed

to a bound on tmix.

We wish to couple the dynamics from the extremal (wired and free) initial

configurations, since arbitrary starting states may induce boundary conditions on

the smaller scales that are not “typical.” Even in the ideal scenario where the

induced boundary conditions have no bridges, though, the probability that we fail

to couple the dynamics from wired and free initial states is at least 1− ε (as per

the RSW estimates). In particular, even in this ideal setting, we could not afford

the O(n2) factor of translating this to a bound on tmix—the actual setting is far

105

worse, replacing the failure probability by 1−n−c (Proposition 3.1.6). An approach

based on the classical block dynamics recursion on the spectral gap (see [67] and

the proofs in [42, 64]) would force one to analyze the dynamics under worst-case

boundary, whereas we would like to restrict attention to the typical boundary

conditions encountered throughout the dynamical process.

Therefore, in Definition 4.2.7 we construct a censored dynamics that mimics

a block dynamics chain, and bound the total variation distance between their

distributions in terms of the probability that we encounter unfavorable boundary

conditions on the sub-blocks before mixing (Proposition 4.2.8). By doing so, we

compare the censored dynamics to the block dynamics with boundary conditions

modified to eliminate all O(log n) bridges over certain edges, paying a cost of nc in

the mixing time. We then let the block dynamics run some nc rounds (paying a

union bound for the probability of ever encountering atypical boundary conditions,

bounded in Corollary 4.2.18). The block dynamics would be making nc many

independent attempts at coupling (possible due to the absence of bridges over

a particular edge) beyond a dual-crossing whose existence has probability n−c/2:

see Lemma 4.2.16. This polynomial bound on the block dynamics coupling time

translates to a quasi-polynomial bound for the censored dynamics via O(log n)

recursions onto smaller scale blocks, yielding the bound on tmix.

Finally, since periodic boundary conditions do not fall in our class of “typical”

boundaries, in Section 4.2.6 we extend this bound first to cylinders, and then to

the torus.

106

Dynamical tools

In this section, we introduce the main techniques we use to control the total

variation distance from stationarity for the FK Glauber dynamics. For the rest of

this chapter, tmix will always be with respect to the FK Glauber dynamics.

4.2.2 Modifications of boundary conditions

Crucial to the proof of Theorem 3 is the modification of boundary bridges so

that we can couple beyond FK interfaces as done in [64]; in this subsection we

define boundary condition modifications and control the effect such modifications

can have on the mixing time.

Definition 4.2.1 (segment modification). Let ξ be an FK boundary condition on

a rectangle Λ which corresponds to a partition P1, ...,Pk of ∂Λ, and let ∆ ⊂ ∂Λ.

The segment modification on ∆, denoted by ξ∆, is the boundary condition that

corresponds to the partition P1 − V (∆), ...,Pk − V (∆) ∪⋃v∈V (∆)v of ∂Λ.

Definition 4.2.2 (bridge modification). Let ξ be an FK boundary condition on

∂Λ, corresponding to a partition P1, . . . ,Pk of ∂Λ. Let Be be the set of disjoint

bridges in ξ∂nΛ over the edge e = (x, y) ∈ ∂nΛ, corresponding to the components

Pij`j=1, as per Definition 3.2.3. The bridge modification of ξ over e, denoted ξe,

is the boundary condition associated to the partition where every Pij is split into

two components,

Pwij

= (v1, v2) ∈ Pij : v1 − x < 0 and Peij

= (v1, v2) ∈ Pij : v1 − x > 0 .

(Observe that, in particular, ξe has no bridges over e.) Define the bridge modification

107

w.r.t. the other sides of ∂Λ analogously.

Definition 4.2.3 (side modification). Let ξ be a boundary condition on ∂Λ,

corresponding to a partition P1, . . . ,Pk of ∂Λ. The side modification ξs is defined

as follows. Split every Pj into its four sides, that is, for i = n, s,e,w, let

P ij = v ∈ Pj : v ∈ ∂iΛ ,

where for the corner vertices the choice is arbitrary (for concreteness, associate the

corner with the side that follows it clockwise). Then for every ξ, the modified ξs

has no components that contain vertices in more than one side of ∂Λ.

Recalling the definition of distances between boundary conditions, we have the

following.

Fact 4.2.4. For a segment ∆, we have d(ξ, ξ∆) ≤ |V (∆)|; for an edge e, we have

d(ξ, ξe) = |Be|; for the side modification ξs, we have that d(ξ, ξs) is bounded above

by three times the number of components in ξ with vertices in multiple sides of ∂Λ.

4.2.3 Censored block dynamics

We next define the censored and systematic block dynamics whose coupling is

the core of the dynamical analysis used to prove Theorem 3. This coupling may

be of general interest in the study of mixing times of monotone Markov chains,

where one only has control on mixing times in the presence of favorable boundary

conditions. We therefore present it in more generality than necessary for the proof

of Theorem 3: consider the heat-bath dynamics for a monotone spin or edge system

on a graph G with boundary ∂G that satisfies the domain Markov property and

has extremal configurations 0, 1 and invariant measure πξG.

108

Definition 4.2.5 (systematic block dynamics). Let B0, . . . , Bs−1 denote a finite

cover of G and for k ≥ 1 let ik := (k − 1) mod s.

The systematic block dynamics (Yk)k≥0 for the FK model is a discrete-time flavor

of the block dynamics w.r.t. Bi, with blocks that are updated in a sequential

deterministic order: at time k, the chain updates E(Bik) by resampling ωE(Bik ) ∼

πξG(· | ωE(G)−E(Bik )).

Remark 4.2.6. The systematic block dynamics as defined has unique invariant

measure πξG, but it is neither time-homogenous nor reversible. If one wanted

a time-homogenous and reversible analogue, one could, e.g., in each time step

update all s blocks sequentially in forward and then reverse order, i.e., in the order

(B0, ..., Bs−1, ..., B0).

Definition 4.2.7 (censored block dynamics). Let B0, ..., Bs−1 be as before and

consider a set Γi of permissible boundary conditions for Bi, and fix ε > 0. The

censored block dynamics (Xt)t≥0 is the continuous-time single-bond (single-site)

heat-bath dynamics that simulates Yk as follows. For a given ε > 0, define

T = T (ε) = maxi

maxξ∈Γi

tξ,Bimix (ε) , (4.2.1)

where tξ,Bimix is the mixing time of standard FK Glauber dynamics on the block Bi

with boundary conditions ξ. Let ik := (k − 1) mod s and let the chain Xt be

obtained from the standard heat-bath dynamics by censoring, as in Theorem 2.2.7,

for every integer k ≥ 1, along the interval ((k − 1)T, kT ], all updates except those

in Bik .

Recall the notation dξω0(t) and the abbreviation Pξω0

(Xt ∈ ·) from §2.2.1.

109

Proposition 4.2.8 (comparison of censored / systematic block dynamics). Let

(Xt)t≥0 and (Yk)k≥0 be the censored and systematic block dynamics, respectively,

w.r.t. some blocks B0, . . . Bs−1 and permissible boundary conditions Γi on G with

boundary conditions ξ and initial state ω0, as per Definitions 4.2.5–4.2.7. Let

ρ := maxk≥1

maxi∈J0,s−1K

Pω0

(Yk∂Bi /∈ Γi

), (4.2.2)

where Yk∂Bi is the boundary condition induced on ∂Bi by the configuration Yk on

G \Bi. Then for every ε > 0, every integer k ≥ 0, and T as in (4.2.1),

∥∥Pω0

(XkT ∈ ·

)− Pω0 (Yk ∈ ·)

∥∥tv≤ k(ρ+ ε) . (4.2.3)

Remark 4.2.9. Although we defined the systematic and censored block dynamics

for deterministic block updates, one could easily formulate the same bound for

the usual block dynamics with random updates, where the s sub-blocks are each

assigned i.i.d. Poisson clocks (cf. [67]), by also randomizing the order in which the

censored block dynamics updates sub-blocks, using the identity coupling on the

corresponding clocks.

Proof of Proposition 4.2.8. We now prove Eq. (4.2.3) by induction on k. Fix

any ω0 and let δk =∥∥Pω0

(XkT ∈ ·

)− Pω0 (Yk ∈ ·)

∥∥tv

denote its left-hand side;

observe that δ0 = 0 by definition, and suppose that δk ≤ k(ρ + ε) for some k.

Denote by i = ik+1 the block that is updated at time k + 1 by the systematic block

dynamics, and let X(i)t and Y

(i)k be the censored and systematic chains corresponding

to the block sequence (B(i+`) mod s)`≥0 (where the block Bi is the first to be updated).

110

By the Markov property and the triangle inequality,

δk+1 ≤1

2

∑ω,ω′

(∣∣∣Pω0(XkT = ω′)− Pω0(Yk = ω′)∣∣∣Pω′(X(i)

T = ω)

+∣∣∣Pω′(X(i)

T = ω)− Pω′(Y (i)1 = ω)

∣∣∣Pω0(Yk = ω′)

)= δk +

∑ω′

Pω0(Yk = ω′)∥∥∥Pω′(X(i)

T ∈ ·)− Pω′(Y (i)1 ∈ ·)

∥∥∥tv. (4.2.4)

The last summand in (4.2.4) satisfies

∑ω

Pω0(Yk = ω)∥∥∥Pω(X

(i)T ∈ ·)− Pω(Y

(i)1 ∈ ·)

∥∥∥tv

≤ Pω0(Yk∂Bi ∈ Γi) maxω:ω∂Bi∈Γi

∥∥∥Pω(X(i)T ∈ ·)− Pω(Y

(i)1 ∈ ·)

∥∥∥tv

+ Pω0(Yk∂Bi /∈ Γi)

≤ (1− ρ)ε+ ρ ,

by the definition of T = T1(ε) and ρ; here we identified the configuration on G−Bi

with the boundary it induces on ∂Bi. Combined with Eq. (4.2.4), this completes

the proof of Eq. (4.2.3).

Remark 4.2.10. In the setting of Proposition 4.2.8, when the initial state is

ω0 ∈ 0, 1 (either minimal or maximal), one can obtain the following improved

bound. Set

T = maxi

maxξ∈Γi

tξ,Bimix (ω0Bi , ε) , (4.2.5)

where tξ,Bimix (ω0, ε) = inft : dξω0(t) ≤ ε, relaxing the previous definition (4.2.1) of T

to only consider the initial state ω0. Let (Xt) be the censored block dynamics w.r.t.

this new value of T , and denote by (X ′t) the modification of (Xt) where, for every

111

k ≥ 1, the configuration of the block Bik (i.e., the block that is to be updated in

the interval ((k − 1)T, kT ]) is reset at time (k − 1)T to the original value of ω0

on that block. We claim that (4.2.3) holds1 for the relaxed value of T in (4.2.5) if

we replace Xt by X ′t. Indeed, all the steps in the above proof of Proposition 4.2.8

remain valid up to the final inequality, at which point the fact that we consider X ′t

(as opposed to Xt) implies that

maxω:ω∂Bi∈Γi

∥∥∥Pω(X(i)T ∈ ·)− Pω(Y

(i)1 ∈ ·)

∥∥∥tv

= maxξ∈Γi

∥∥∥Pω0Bi(X

(i)T ∈ ·)− π

ξBi

∥∥∥tv,

which is at most ε when T is as defined in (4.2.5).

Proof of main result

In what follows, we prove Theorem 3 by combining the equilibrium estimates

of §3.2 with the dynamical tools provided in §4.2.1. We first establish an analog of

Theorem 3 (Theorem 4.2.14) for “typical” boundary conditions (defined in §4.2.4

below), and then, using Proposition 3.2.13, derive from it the case of periodic

boundary conditions in §4.2.6. The effect of boundary bridges (which may foil the

multiscale coupling approach, as described in §4.2.1) is controlled by restricting

the analysis to those boundary conditions that have O(log n) bridges, and applying

Proposition 4.2.8 to bound the mixing time under such boundary conditions. We

now define the favorable boundary conditions for which we prove a mixing time

upper bound of nO(logn).

1In fact, (4.2.3) is valid for X ′t with the relaxed T in (4.2.5) for every ω0, not just for the maximal

and minimal configurations; however, it is when ω0 ∈ 0, 1 that the modified dynamics X ′t can

easily be compared to Xt, and thereafter to Xt, via the censoring inequality of Theorem 2.2.7.

112

4.2.4 Typical boundary conditions

We define the class of “typical” boundary conditions on a segment (e.g., ∂nΛ).

Definition 4.2.11 (typical boundary conditions on a segment). For K > 0, N ≥ 1,

and a segment L, let ΞK,N be the set of boundary conditions ξ on L such that

|Be(ξ)| ≤ K logN for every e ∈ L .

We will later see (as a consequence of Lemma 4.2.17 below) that the boundary

conditions on each of the sides of a box Λ induced by the infinite-volume FK measure

πZ2 belong to the class of “typical” boundary conditions with high probability.

Next, we define the global property we require of typical boundary conditions.

Definition 4.2.12 (typical boundary conditions on ∂Λ). Let ΥK1,K2,N = ΥΛK1,K2,N

be the set of boundary conditions ξ on ∂Λ such that ξ∂iΛ ∈ ΞK1,N for every

i = n, s,e,w, and ξ has at most K2 logN distinct components with vertices on

different sides of ∂Λ.

Remark 4.2.13. The wired and free boundary conditions on a side ∂iΛ are always

in ΥK1,K2,N whenever K1 logN ≥ 1 and K2 logN ≥ 1 (in the former all vertices are

in just one component and in the latter no two vertices are in the same component).

4.2.5 Mixing under typical boundary conditions

Since periodic boundary conditions are not in ΥK1,K2,N for any K2 > 0, we first

bound the mixing time on rectangles ΛN,N ′ where N ′ = bαNc for α ∈ (0, 1], with

boundary conditions ξ ∈ ΥK1,K2,N .

113

Theorem 4.2.14. Let q ∈ (1, 4] and fix α ∈ (0, 1] and K1, K2 > 0. Consider the

Glauber dynamics for the critical FK model on ΛN,N ′ with αN ≤ N ′ ≤ N and

boundary conditions ξ ∈ ΥK1,K2,N . Then there exists c = c(α, q,K1, K2) > 0 such

that

tmix . N c logN .

Observe that if we define

ΥK,N := ΥK,2K,N , (4.2.6)

clearly ΥK1,K2,N ⊂ ΥmaxK1,K2,N , so it suffices to consider ΥK,N for general K > 0.

The proof of Theorem 4.2.14 proceeds by analyzing the censored and systematic

block dynamics on Λ, obtaining good control on the systematic block dynamics

using the RSW estimates of [34], then comparing it to the censored block dynamics.

The choice of parameters for which we will apply Proposition 4.2.8 is the following.

Definition 4.2.15 (block choice for censored / systematic block dynamics). Let

q ∈ (1, 4] and for any n′ ≤ n ≤ N , consider the critical FK Glauber dynamics on

Λn,n′ . Let

Be = Jn4, nK× J0, n′K ,

Bw = J0, 3n4

K× J0, n′K ,

ordered as B0 = Be, B1 = Bw as in the setup of Proposition 4.2.8. For K =

maxK1, K2 given by Theorem 4.2.14, let Γi = ΥK,N be the set of permissible

boundary conditions for the block Bi in Λn,n′ .

Before proving Theorem 4.2.14 we will prove two lemmas that will be necessary

114

for the application of Proposition 4.2.8. We first introduce some preliminary

notation.

For any n ≤ N , label the following edges in ∂Λn,n′ :

e?s = (bn2c+ 1

2, 0) , and e?n = (bn

2c+ 1

2, n′) .

Recall the definitions of the bridge modification ξe and the side modification ξs

from Definitions 4.2.2–4.2.3. We will, throughout the proof of Theorem 4.2.14, for

any boundary condition ξ on ∂Λn,n′ , let the modification ξ′ ≤ ξ be given by

ξ′ := ξe?s ∧ ξe?n ∧ ξs , (4.2.7)

i.e., the bridge modification of ξ on e?s and e?n, combined with the side modification ξs.

If ΞK,N ,ΥK,N are the sets of boundary conditions defined in Definition 4.2.15,

we let Ξ′K,N ,Υ′K,N be the sets corresponding to the modification ξ 7→ ξ′ of every

element in the original sets. Observe that Υ′K,N ⊂ ΥK,N and likewise, Ξ′K,N ⊂ ΞK,N .

Lemma 4.2.16. Let α ∈ (0, 1] and consider the systematic block dynamics Ykk∈N

on Λn,n′ with bαnc ≤ n′ ≤ n and blocks given by Definition 4.2.15. There exist

cY , c?(α, q) > 0 such that for every two initial configurations ω1, ω2, and every

boundary condition ξ on ∂Λn,n′, modified to ξ′ by Eq. (4.2.7), for all k ≥ 2,

‖Pξ′ω1(Yk ∈ ·)− Pξ′ω2

(Yk ∈ ·)‖tv ≤ exp(−cY kn−c?) .

In particular, for all k ≥ 2,

maxω0

‖Pξ′ω0(Yk ∈ ·)− πξ

′

Λn,n′‖tv ≤ exp(−cY kn−c?) .

115

n4

n2

3n4

ξ′

1/0 Be

Figure 4.1: If the depicted dual-crossing exists under any (ξ′, ωi), and the bridgesover e∗s , e

∗n are disconnected, one can couple the two chains on the green shaded

region, and in particular on Be −Bw.

Proof. We construct a coupling between the two systematic block dynamics chains,

starting from two arbitrary initial configurations ω1, ω2, as follows. The systematic

block dynamics first samples a configuration on Be (the interior of Be) according to

πξ′,ωiBe

for i = 1, 2, where (ξ′, ωi) is the boundary condition induced by ωiBw−Be∪ ξ′

on ∂Be. By Proposition 3.1.6, applied to the box

R∗ = B∗w ∩B∗e ,

and monotonicity in boundary conditions,

π1Be

(e?sR∗←→ e?n) & n−c? ,

where c?(min12, α, ε = 1

4, q) > 0 is given by that proposition.

116

We can condition on the west-most vertical dual-crossing between e?s and

e?n (if such a dual-crossing exists) as follows: reveal the open components of

∂Be ∩ J0, bn2cK× J0, n′K, so that no edges in other components are revealed (as in

e.g., steps (1)–(4)). If the open components do not connect to the eastern half of

∂R∗, i.e., to ∂R∗ ∩ Jbn2c+ 1, nK× J0, n′K, then it must be the case that the desired

open dual-crossing exists and can be exposed without revealing any information

about edges east of it.

By monotonicity in boundary conditions, if under π1Be

a vertical dual-connection

from e?s to e?n exists, the grand coupling (see §2.2.1) ensures that the same under

πξ′,ωiBe

for any ωiΛn,n′−Be. By definition of the modification ξ′, there are no bridges

over e?s , no bridges over e?n, and no components of ξ′ with vertices in multiple sides

of ∂Λn,n′ ; thus, conditional on this vertical dual-crossing, the following event holds:

⋂v∈ ∂R∩J0, n

2K×J0,n′K

w∈ ∂R∩Jn2,nK×J0,n′K

ω : v←→6 w through ω and connections in ξ′ .

By the domain Markov property (see Fig. 4.1), for any pair ω1Bw−Beand ω2Bw−Be

,

πξ′,ω1

Be

(ωJ 3n

4,nK×J0,n′K

∣∣ e?s R∗←→ e?n

)d= πξ

′,ω2

Be

(ωJ 3n

4,nK×J0,n′K

∣∣ e?s R∗←→ e?n

),

using that the boundary conditions to the east of the vertical dual-crossing are the

same under both measures. (In the presence of bridges over e?s or e?n the above

distributional equality does not hold; different configurations west of such a dual-

crossing could still induce different boundary conditions east of the dual-crossing,

preventing coupling (as illustrated in Fig. 1.5)—cf. the case of integer q where this

problem does not arise thanks to the event Eζ .)

117

This implies that, on the event e?sR∗←→ e?n, the grand coupling couples the two

systematic block dynamics chains so that they agree on E(Λn,n′) − E(Bw) with

probability 1. In this case, let η be the resulting configuration on E(Be)− E(Bw),

so that

η = Y1E(Be)−E(Bw) .

If the two chains were coupled on E(Be)− E(Bw), the boundary conditions (ξ′, η)

induced on ∂Bw would be the same for any pair of systematic block dynamics

chains with initial configurations ω1, ω2; in particular the identity coupling would

couple them on all of Λn,n′ in the next step when Bw is resampled from πξ′,ηBw

. Thus,

for some c > 0,

‖Pξ′ω1(Y2 ∈ ·)− Pξ′ω2

(Y2 ∈ ·)‖tv ≤ 1− cn−c? .

Since the systematic block dynamics is Markovian and all of the above estimates

were uniform in ω1 and ω2, the probability of not having coupled in time k under

the grand coupling is bounded above by

(1− cn−c?)bk/2c ≤ exp(−cbk/2cn−c?) .

The next lemma will be key to obtaining the desired upper bound on ρ as

defined in (4.2.2); it shows that with high probability, the boundary conditions

induced by the FK measure on a segment will be in ΞK,N , hence the term “typical”

boundary conditions.

Lemma 4.2.17. Fix q ∈ (1, 4]. There exists cΥ(q) > 0 so that, for every ΞK,N

given by Definition 4.2.11 on Λn,n′ with n′ ≤ n ≤ N and K > 0, and every boundary

118

condition ξ,

πξBe(ω∂eBw

/∈ ΞK,N) . N−cΥK ,

where ω∂eBwdenotes the boundary conditions induced on ∂eBw by ωE(Be)−E(Bw)∪ξ.

The same statement holds when exchanging e and w.

Proof. By symmetry, it suffices to prove the bound for the boundary conditions on

∂eBw. Consider the rectangle

R = Jn2, nK× J0, n′K .

By Proposition 3.2.8 with aspect ratio 12, there exists c(q) = c(α = 1

2, q) > 0 such

that, for every edge e ∈ ∂eBw and every boundary condition η on ∂R,

πηR(|Be| ≥ K logN) . N−cK ,

where, for a configuration ωR on R, we recall that |Be| is the number of disjoint

bridges in ωRR−Bw∪ ξR over e. A union bound over all n′ edges on ∂eBw implies

that

maxηπηR(ω∂eBw

/∈ ΞK,N) . n′N−cK . N−cK+1 ,

using n ≤ N . Consequently,

πξBe(ω∂eBw

/∈ ΞK,N) = EπξBe

[πξRR (ω∂eBw

/∈ ΞK,N)]. N−cK+1 ,

where the expectation is w.r.t. πξBeover the boundary conditions ξR induced on R

by ξ and the configuration on Be −R. This concludes the proof of the lemma.

Corollary 4.2.18. Fix q ∈ (1, 4], and consider the systematic block dynamics on

119

Λn,n′ for n′ ≤ n ≤ N with block choices as given in Definition 4.2.15. There exists

cΥ(q) > 0 so that, for every fixed K > 0 and every boundary condition ξ′ ∈ Υ′K,N

on ∂Λn,n′,

ρ . N−cΥK ,

where ρ is as defined as in (4.2.2) w.r.t. the initial configuration ω0 ∈ 0, 1 and

the permissible boundary conditions ΥK,N .

Proof. Let Yk be the systematic block dynamics on Λn,n′ where n ≤ N . Recall the

definition of ρ in Eq. (4.2.2), so that in the present setting,

ρ = maxω0∈0,1

maxk≥1

maxi∈e,w

Pξ′ω0(Yk∂Bi /∈ ΥK,N) .

In the first time step, ω0Beinduces wired or free boundary conditions on ∂wBe

and so, by Remark 4.2.13, the boundary condition on ∂wBe is trivially in ΞK,N .

Furthermore, the boundary conditions on ∂n,e,sBe also belong to ΞK,N by the

hypothesis ξ′ ∈ ΥK,N . Finally, there cannot be more than 2K logN components

in the boundary condition on ∂Be consisting of vertices on multiple sides for the

following reason: as a result of the side modification on ξ′, such components can

only arise from connections between ∂wBe and the bridges in B(n/4,0) and B(n/4,n′);

however, there are at most K logN bridges in each set under any configuration

on Λ− Be (summing to at most 2K logN components, as claimed). Altogether,

Y1∂Be∈ ΥK,N deterministically.

To address all subsequent time steps, by reflection symmetry and the definition

of the systematic block dynamics, is suffices to consider Y2∂Bw. By Lemma 4.2.17,

the probability that a boundary condition on ∂eBw induced by the systematic

dynamics will not be in ΞK,N is O(N−cΥK), with cΥ > 0 from that lemma. The fact

120

that, deterministically, the boundary conditions on ∂n,s,wBw are in ΞK,N , and there

are at most 2K logN components of the boundary condition on ∂Bw containing

vertices of multiple sides of ∂Bw, follows by the same reasoning argued for the first

time step.

We are now in a position to prove Theorem 4.2.14.

Proof of Theorem 4.2.14. Consider Λ = ΛN,N ′ with aspect ratio α ∈ (0, 1] and

boundary conditions ξ ∈ ΥK,N for a fixed

K ≥ K0 := 6(c? + 1) maxc−1Υ , 1 , (4.2.8)

where c? = c?(minα, 12, 1

4, q) is the constant given by Proposition 3.1.6, and

cΥ = cΥ(q) is given by Corollary 4.2.18. It suffices to prove the proposition for all

K sufficiently large, as ΥK,N ⊂ ΥK′,N for every K ≤ K ′.

We prove the following inductively in n ∈ J1, NK: for every K > K0 as above,

every (α ∧ 12)n ≤ n′ ≤ n, and every ξ ∈ ΥK,N , if

tn = N2(c?+λ+1) log4/3 n where λ := 32K log q + 5 ,

then Glauber dynamics for the critical FK model on Λn,n′ has

‖Pξ1(Xtn ∈ ·)− Pξ0(Xtn ∈ ·)‖tv ≤ N−3 . (4.2.9)

To see that Eq. (4.2.9) implies Theorem 4.2.14, note that (2.2.7), with the choice

n = N , implies that dtv(N c(α,q) logN) = O(1/N) = o(1) for some c(α, q) > 0.

For the base case, fix a large constant M , where clearly tmix = O(1) for all n ≤M .

121

Next, let m ∈ JM,NK, and assume (4.2.9) holds for all n ∈ J1,m− 1K. Consider

the censored and systematic block dynamics, (Xt)t≥0 and Ykk≥0, respectively, on

the blocks defined in Definition 4.2.15 on Λm = Λm,m′ for some (α∧ 12)m ≤ m′ ≤ m

and boundary conditions ξ ∈ ΥK,N .

Recall that ξ ∈ ΥK,N has at most K logN bridges over any edge and at most

2K logN components spanning multiple sides of ∂Λm; thus, by Fact 4.2.4, the

boundary modification ξ′ defined in (4.2.7) satisfies d(ξ′, ξ) ≤ 8K logN . By the

definition of λ, we have |E|2q4d(ξ′,ξ) = o(Nλ). Hence, by Lemma 2.2.13 (Eq. (2.2.11),

where we increased the time on the right-hand to Nλ, for large enough N , by the

monotonicity of dtv) and the above bound on d(ξ′, ξ), we have that for all k, T ≥ 0,

‖Pξ1(XNλkT ∈ ·)− Pξ0(XNλkT ∈ ·)‖tv

≤ 2 maxω0∈0,1

‖Pξω0(XNλkT ∈ ·)− πξΛm‖tv

≤ 16 maxω0∈0,1

‖Pξ′ω0(XkT ∈ ·)− πξ

′

Λm‖tv + 2e−N

λ/4

,

and subsequently, by Theorem 2.2.7,

‖Pξ1(XNλkT ∈ ·)−Pξ0(XNλkT ∈ ·)‖tv ≤ 16 maxω0∈0,1

‖Pξ′ω0(XkT ∈ ·)−πξ

′

Λm‖tv+2e−N

λ/4

.

(4.2.10)

We will next show that the first term in the right-hand above satisfies

maxω0∈0,1

‖Pξ′ω0(XkT ∈ ·)− πξ

′

Λm‖tv = o(N−3) , (4.2.11)

which will imply (4.2.9) if we choose k, T such that NλkT ≤ tm. By triangle

122

inequality,

maxω0∈0,1

‖Pξ′ω0(XkT ∈ ·)− πξ

′

Λm‖tv

≤ maxω0∈0,1

‖Pξ′ω0(XkT ∈ ·)− Pξ′ω0

(Yk ∈ ·)‖tv + maxω0∈0,1

‖Pξ′ω0(Yk ∈ ·)− πξ

′

Λm‖tv

≤ maxω0∈0,1

‖Pξ′ω0(XkT ∈ ·)− Pξ′ω0

(Yk ∈ ·)‖tv + e−cY km−c?

,

where the last inequality is valid for every k ≥ 2 by Lemma 4.2.16. Using Υ′K,N ⊂

ΥK,N and Proposition 4.2.8,

maxω0∈0,1

‖Pξ′ω0(XkT ∈ ·)− πξ

′

Λm‖tv ≤ k(ρ+ ε(T )) + e−cY km

−c?,

and so, combined with (4.2.10),

‖Pξ1(XNλkT ∈ ·)− Pξ0(XNλkT ∈ ·)‖tv ≤ 16k(ρ+ ε(T )) + 16e−cY km−c?

+ 2e−Nλ/4

,

(4.2.12)

where ρ and ε were given in (4.2.1)–(4.2.2), that is, in our context,

ε(T ) = maxω′∈Ω

maxi∈e,w

maxζ∈Υ

BiK,N

‖Pζ,Biω′ (XT ∈ ·)− πζBi‖tv ,

ρ = maxk≥1

maxi∈e,w

Pω0

(Yk∂Bi /∈ ΥBi

K,N

).

We will bound ε(T ) by the inductive assumption for the choice of

T := ktb3m/4cNλK logN , where k := c−1

Y (c? + 6)N c? logN . (4.2.13)

In order to apply the induction hypothesis for a box whose side lengths are smaller

123

by a constant factor vs. the original dimensions of m ×m′, we repeat the above

analysis for the sub-block Bi (whose dimensions are b34mc × m′), and get from

Eq. (2.2.7) and the above arguments that

ε(T ) . N2 maxi∈e,w

maxζ∈Υ

BiK,N

‖Pζ,Bi0 (XT ∈ ·)− Pζ,Bi1 (XT ∈ ·)‖tv ,

which by reapplying (4.2.12) at the lower scale of the Bi’s implies that

ε(T ) . N2k(ρ′ + ε′( T

kNλ ))

+N2e−cY km−c?

+N2e−Nλ/4

,

where ε′(T ) and ρ′ are the counterparts of ε(T ) and ρ w.r.t. the sub-blocks (as per

Definition 4.2.15) of Bi rotated by π/2. (N.b. this rotation is crucial to ensuring

that the aspect ratios of the rectangles we consider remain uniformly bounded as we

recurse down in scale, and consequently the coupling probabilities satisfy the same

lower bound; this rotation is also what forces us to maintain “typical” boundary

conditions on all four sides of the rectangles we are considering as opposed to, say,

just on ∂e,wΛ.)

This yields the following new bound on (4.2.11):

maxω0∈0,1

‖Pξ′ω0(XkT ∈ ·)− πξ

′

Λm‖tv

. N2k2(ρ+ ρ′ + ε′( T

kNλ ))

+ kN2e−cY km−c?

+ o(N−3).

Note that the dimensions of the sub-blocks of Bi (those under consideration in

ε′(T )) are b34mc × b3

4m′c. Hence, by the inductive assumption at scale b3

4mc and

Eq. (2.2.7),

ε′(tb3m/4c

)= O(1/N) ,

124

which, along with (2.2.7) and the sub-multiplicativity of dtv(t), yields that for T

from (4.2.13),

ε′( TkNλ ) = ε′

(tb3m/4cK logN

). N−K ≤ N−6(c?+1) .

By Corollary 4.2.18, we have ρ . N−cΥK ≤ N−6(c?+1) by our choice of K0,

and similarly for ρ′. So, for k = N c?+o(1) as in (4.2.13), k2ρ . N−4c?−6+o(1) =

o(N−5), and similarly, k2ρ′ = o(N−5). Finally, this choice of k guarantees that

kN2 exp(−cY km−c?) is at most kN−c?−4 = o(N−3). Combining the last three

displays with these bounds yields (4.2.11). The proof is concluded by noting that

indeed NλkT ≤ N2c?+2λ+o(1)tb3m/4c ≤ tm.

4.2.6 Mixing on the torus

Here we extend Theorem 4.2.14 to the n×n torus, proving Theorem 3. Observe

that the periodic FK boundary conditions identified with (Z/nZ)2 in fact have

order n components with vertices on multiple sides of ∂Λ. We thus have to extend

the bound of Theorem 4.2.14 to periodic boundary conditions using the topological

structure of (Z/nZ)2. The proof draws from the extension of mixing time bounds

in [64] from fixed boundary conditions to (Z/nZ)2. In the present setting, having

to deal with a specific class of boundary conditions forces us to reapply the bridge

modification and the censored and systematic block dynamics techniques.

We first bound the mixing time on a cylinder with typical boundary conditions

125

on its non-periodic sides. In what follows, for any Λn,n′ , label the following edges:

e?sw = (0, bn′2c+ 1

2) , e?se = (n, bn′

2c+ 1

2) ,

e?nw = (0, b9n′

10c+ 1

2) , e?ne = (n, b9n′

10c+ 1

2) .

Then define the modification ξ′ of boundary conditions ξ by

ξ′ = ξe?sw ∧ ξe?se ∧ ξe?nw ∧ ξe?ne ∧ ξs (4.2.14)

and define Ξ′K,N ,Υ′K,N as before, for the new modification. We say that a boundary

condition on ∂n,sΛ is in ΥK,N if its restriction to each side is in ΞK,N and there

are fewer than 2K logN distinct components with vertices in ∂nΛ and ∂sΛ, and

analogously for boundary conditions on ∂e,wΛ.

Theorem 4.2.19 (Mixing time on a cylinder). Fix q ∈ (1, 4], α ∈ (0, 1] and K > 0.

There exists some c(α, q,K) > 0 such that the critical FK model on Λ = ΛN,N ′ with

αN ≤ N ′ ≤ α−1N and boundary conditions, denoted by (p, ξ), that are periodic on

∂n,sΛ and ξ ∈ ΥK,N on ∂e,wΛ, satisfies tmix . N c logN .

Proof. We will use a similar approach as in the proof of Theorem 4.2.14 to reduce

the cylinder to rectangles with “typical” boundary conditions. It suffices to prove

the theorem for large K, since ΥK,N ⊂ ΥK′,N for K ≤ K ′. We establish it for every

fixed

K ≥ K0 +K ′0 where K0 = 4(c? + 1)(c−1Υ ∨ 1) and K ′0 = K0(c−1

Ψ ∨ 1) ,

in which c? = c?(α5, 1

4, q) > 0 is given by Proposition 3.1.6, the constant cΥ is

c(2α5, q) > 0 from Proposition 3.2.8, and cΨ = cΨ(3α

5, q) > 0 is given by Proposi-

126

B0

B1

R

1/0

ξ′ ξ′

Figure 4.2: Left and center: block choices B0, B1 for the censored and systematicblock dynamics block dynamics chain on ΛN,N ′ with periodic boundary on ∂n,sΛ,and ξ′ on ∂e,wΛ. Right: the dual crossings in R∗s and R∗N which allow coupling onthe set R.

tion 3.2.13.

Define, as in Definition 4.2.7, the censored and systematic block dynamics on

B0 := J0, NK× J0, N ′5

K ∪ J0, NK× J2N ′

5, N ′K ,

B1 := J0, NK× J0, 3N ′

5K ∪ J0, NK× J4N ′

5, N ′K .

The choice of boundary class on Bi for i = 0, 1 is Γi = Υ3K,N . Observe that

by translating vertically on the universal cover, the blocks B0 and B1 are, by

construction, N × 45N ′ rectangles with non-periodic boundary conditions. These

blocks and the coupling scheme are depicted in Figure 4.2.

It again suffices, by Eq. (2.2.7), to show that there exists some c(α, q,K) > 0

such that

‖Pp,ξ1 (XNc logN ∈ ·)− Pp,ξ0 (XNc logN ∈ ·)‖tv ≤ N−3 . (4.2.15)

127

In the setting of the cylinder, the side modification (p, ξs) of (p, ξ) only disconnects

∂eΛ from ∂wΛ, and so, if ξ′ is as in (4.2.14), then d(ξ′, ξ) ≤ 6K logN . Thus,

by (2.2.11), the triangle inequality and Theorem 2.2.7 (as explained in the derivation

of (4.2.10)), if

tN = NλkT for λ := 24K + 5

(so that |E|2q4d(ξ′,ξ) = o(Nλ)), then for every k, T ≥ 0,

‖Pp,ξ1 (XtN ∈ ·)− Pp,ξ0 (XtN ∈ ·)‖tv

≤ 16 maxω0∈0,1

‖Pp,ξ′ω0(XkT ∈ ·)− πp,ξ

′

Λ ‖tv + 2e−Nλ/4

,

which, by Proposition 4.2.8, is at most

16 maxω0∈0,1

‖Pp,ξ′ω0(Yk ∈ ·)− πp,ξ

′

Λ ‖tv + 16k(ρ+ ε(T )) + 2e−Nλ/4

, (4.2.16)

where ε(T ) and ρ are given by (4.2.1) and (4.2.2), respectively, w.r.t. the blocks

B0, B1, the permissible boundary conditions Υ3K,N , and the initial configuration

ω0 ∈ 0, 1:

ε(T ) = maxω′∈Ω

maxi∈0,1

maxζ∈Υ

Bi3K,N

‖Pζ,Biω′ (XT ∈ ·)− πζBi‖tv ,

ρ = maxk≥1

maxi∈0,1

Pω0

(Yk∂Bi /∈ ΥBi

3K,N

).

We next bound the first term in the right-hand side of Eq. (4.2.16) by the

probability of not coupling the systematic block dynamics chains started from two

arbitrary initial configurations under the grand coupling (cf. Lemma 4.2.16). In

128

the first time step, we try to couple the chains started from ω1, ω2 on

R := J0, NK× J3N ′

5, 4N ′

5K ,

so that in the second step the identity coupling couples them on all of Λ. It suffices

to couple the systematic chains started from ω1 = 0 and ω2 = 1 under the grand

coupling. In order to couple samples from the (0, ξ′) and (1, ξ′) boundary conditions

on R (induced by ω1 = 0 and ω2 = 1 resp.), define the following two sub-blocks of

B0:

Rs := J0, NK× J2N ′

5, 3N ′

5K , Rn := J0, NK× J4N ′

5, N ′K .

By Proposition 3.1.6, monotonicity in boundary conditions, and the FKG inequality,

minηπη,ξ

′

B0

(e?sw

R∗s←→ e?se , e?nw

R∗n←→ e?ne

)& N−2c? .

By the Domain Markov property, and the definition of the boundary modification

ξ′,

π1,ξ′

B0

(ωR

∣∣ e?sw R∗s←→ e?se , e?nw

R∗n←→ e?ne

)d= π0,ξ′

B0

(ωR

∣∣ e?sw R∗s←→ e?se , e?nw

R∗n←→ e?ne

).

As before, using the grand coupling and revealing edges from ∂sRs and ∂nRn until

we reveal a pair of such horizontal dual-crossings, by monotonicity, we can couple

πω1,ξ′

B0and πω2,ξ′

B0on R with probability at least cN−2c? . On that event, the two

chains are coupled in the next step (and thereafter) on all of Λ with probability

1. By the definition of the systematic block dynamics, we conclude that, for some

129

cY > 0 and every k ≥ 2,

maxω0∈0,1

‖Pp,ξ′ω0(Yk ∈ ·)− πp,ξ

′

Λ ‖tv ≤ exp(−cY kN−2c?) .

To bound ρ, first note that, for ω0 ∈ 0, 1, the block B0 has boundary conditions

(0, ξ′) or (1, ξ′), both of which are in Υ3K,N by Remark 4.2.13. Thereafter, the

uniformity of Proposition 3.2.8 in boundary conditions implies that for every η,

πη,ξ′

B0(ω∂B1

/∈ ΞK,N) . N−cΥK ,

and likewise when exchanging B0 and B1. We need to also bound the number of

boundary components intersecting distinct sides of B0 or B1. We can bound the

connections between ∂n,sBi and ∂e,wBi (for i = 0, 1) deterministically by 4K logN as

in the proof of Corollary 4.2.18. In the present setting there could also be (multiple)

open components connecting ∂sBi to ∂nBi in Λ− Bi. By Proposition 3.2.13 and

monotonicity in boundary conditions, for every η,

πη,ξB0(|ΨΛ−B1| ≥ K logN) . N−cΨK ,

where, as in that proposition, |ΨΛ−B1| is the number of distinct vertical crossings

of Λ−B1. By the choices of K0 and K ′0, a union bound yields

ρ . maxηπη,ξ

′

B0(ω∂B1

/∈ Υ3K,N) . N−4c?−4 .

Observe that on their respective translates, B0 and B1 are N × 45N ′ rectangles,

so we can bound maxi maxξ∈Υ3K,Ntξ,Bimix using Theorem 4.2.14; by that theorem,

130

rotational symmetry, and the sub-multiplicativity of dtv, we have that for some

cB(α, q,K) > 0,

ε(T ) . exp(−c−1B TN−cB logN) ,

uniformly over αN ≤ N ′ ≤ α−1N . Altogether, combining the bounds on ρ, ε, and

the systematic block dynamics distance from stationarity, in Eq. (4.2.16), we see

that for

k = N2c?+1 and T = N (cB+1) logN

one has

‖Pp,ξ1 (XtN ∈ ·)− Pp,ξ0 (XtN ∈ ·)‖tv = o(N−3) ,

implying Eq. (4.2.15) and concluding the proof.

Proof of Theorem 3. The theorem is obtained by reducing the mixing time on

the torus to that on a cylinder and then applying Theorem 4.2.19. Fix α ∈ (0, 1] and

consider Λ = Λn,n′ with αn ≤ n′ ≤ α−1n and periodic boundary conditions, denoted

by (p), identified with (Z/nZ)× (Z/n′Z) (the special case n′ = n is formulated in

Theorem 3).

Let c? = c?(α5, 1

4, q) > 0 be given by Proposition 3.1.6 and let cΥ,cΨ, K0 and K ′0

be given as in the proof of Theorem 4.2.19. Define K = K0 +K ′0. We consider the

censored and systematic block dynamics with the block choices,

B0 := J0, nK× J0, n′5K ∪ J0, nK× J2n′

5, n′K and

B1 := J0, nK× J0, 3n′

5K ∪ J0, nK× J4n′

5, n′K ,

131

and boundary class

Υp3K,n :=

ξ : ξ∂n,sΛ = p, ξ∂e,wΛ ∈ Υ3K,n

.

By Theorem 2.2.7, the triangle inequality and Proposition 4.2.8, for every k, T ≥ 0,

‖Pp1(XkT ∈ ·)− Pp0(XkT ∈ ·)‖tv ≤ 2 maxω0∈0,1

‖Ppω0(Yk ∈ ·)− πpΛ‖tv + 2k(ρ+ ε(T )) ,

where ρ and ε(T ) are w.r.t. the class Υp3K,n of permissible boundary conditions. It

suffices, as before, to prove that the right-hand side is o(n−3) and then use (2.2.7)

and the sub-multiplicativity of dtv(t) to obtain the desired result.

Recall the edges e?sw, e?nw, e

?se and e?ne on Λn,n′ . As in the proof of Theorem 4.2.19,

if

Rs := J0, nK× J2n′

5, 3n′

5K , Rn := J0, nK× J4n′

5, n′K ,

then by Proposition 3.1.6 and the FKG inequality, we have

π1,pB0

(e?sw

R∗s←→ e?se , e?nw

R∗n←→ e?ne

)& n−2c? .

Crucially, while no boundary modification was done in this case, the periodic

sides of B0 have no bridges over the four designated edges, and the two horizontal

dual-crossings, from the event above, disconnect its non-periodic sides (∂nB0 and

∂sB0) from ∂nB1 and ∂sB1. Therefore, if that event occurs for the systematic

block dynamics chain started from ω0 = 1, the grand coupling carries it to the

chains started from all other initial states, and yields a coupling of all these chains

132

on J3n5, 4n

5K × J0, n′K ⊃ ∂B1. By definition of the systematic block dynamics and

sub-multiplicativity of dtv(t), for k ≥ 2,

maxω0∈0,1

‖Ppω0(Yk ∈ ·)− πpΛ‖tv ≤ exp(−cY kn−2c?) . (4.2.17)

Observe that at every time step of the systematic block dynamics, the block Bi

(i = 0, 1) is an n× 45n′ rectangle with periodic boundary conditions on ∂e,wBi and

boundary conditions η induced by the chain on ∂n,sBi. By Theorem 4.2.19, for

some c(α, q,K) > 0,

maxi

max(p,η)∈Υp3K,n

t(p,η),Bimix . nc logn ,

and by sub-multiplicativity of dtv(t), we have ε(T ) . exp(−c−1Tnc logn). As in

the proof of Theorem 4.2.19, since the estimate on ρ was uniform in the boundary

conditions, we again have ρ . n−4c?−4 (using Propositions 3.2.8 and 3.2.13). Com-

bining the bounds on ρ and ε with (4.2.17), there exists some c(α, q,K) > 0 such

that

‖Pp1(Xnc logn ∈ ·)− Pp0(Xnc logn ∈ ·)‖tv = o(n−3) ,

as required.

133

Chapter 5

Slow mixing in phase coexistence

regions

In this chapter, we prove Theorems 2 and 4. In particular, we show that as soon

as there are multiple distinct infinite volume Potts or FK measures, their respective

Glauber dynamics on (Z/nZ)2 are exponentially slow. The previous works [12, 13]

had used complicated geometric bottlenecks and the perturbative Pirogov–Sinai

theory [77, 78] to establish exponentially slow mixing for these dynamics in regimes

where one of q or β is sufficiently large on (Z/nZ)d for general d ≥ 2. Our proofs in

this section hold only for d = 2, but use a simple geometric bottleneck consisting of

three vertical and three horizontal loops around the torus, for which we can prove

exponentially slow mixing as soon as there is an exponential decay of connectivities

under π1Z2 (without relying on perturbative techniques).

Let us first sketch the key idea behind the proofs, then prove slow mixing in

phase coexistence regimes for the Potts Glauber dynamics in Section 5.1, and for

Swendsen–Wang and FK Glauber dynamics more generally, in Section 5.2.

134

For completeness, in Section 5.3, we prove that for every q ≥ 2 and β > βc(q),

the Potts Glauber dynamics on Λn with free boundary conditions is slow.

5.0.1 Proof idea

A standard approach to mixing time lower bounds is to construct a bottleneck

set of configurations which has a very small conductance (2.2.14), as this upper

bounds the spectral gap per Cheeger’s inequality (2.2.15). Suppose there are

multiple infinite-volume Gibbs measures, say corresponding to two distinct colors

in the Potts case, and wired and free in the FK case. We establish a bottleneck

in the state space consisting of configurations containing vertical and horizontal

crossings (either of monochromatic vertices, or of open edges) each forming a loop

around the torus. The basic idea is that a combination of a horizontal loop and a

vertical loop in the torus can be translated on the universal cover of the torus to

form a macroscopic wired FK circuit. Escaping such configurations via a pivotal

edge would require a macroscopic dual-crossing inside the circuit, an event which,

for any given circuit, has an exponentially small probability (see Theorem 2.1.2).

Unfortunately, conditioning on the locations of these two loops includes negative

information about the interior of the circuit and prevents us from appealing to the

decay of correlations estimates. If we instead considered two pairs of horizontal and

vertical loops: one could expose the required wired circuit with no information on

its interior. However, a subtler problem then arises, where after exposing one pair

of loops (say the vertical ones), revealing the second (horizontal) pair might leave

the potentially pivotal edge outside of the formed wired circuit, preventing us from

estimating the probability of it being pivotal. It turns out that using three pairs

of horizontal and vertical loops supports a suitable protocol for exposing a wired

135

circuit such that the potential edge is pivotal only if it supports a macroscopic

dual-crossing within that circuit, thus leading to the desired lower bound.

5.1 Potts Glauber dynamics on (Z/nZ)2 in phase

coexistence regions

In this section, we will prove that as soon as there are multiple distinct infinite-

volume Potts measures on Z2, the corresponding Potts Glauber dynamics is expo-

nentially slow on (Z/nZ)2. We use in particular, the fact that for a given (p, q),

there are multiple Z2-measures if and only if π1Z2,p,q has an infinite cluster; in turn,

the planarity of Z2 and results of [5, 34] ensure that whenever π1Z2,p,q has an infinite

cluster, its dual-clusters are almost surely finite and have exponential tails. Namely,

by the dual version of (2.1.1) when q ≥ 1 and p > pc(q), and by Theorem 2.1.2

when q > 4 and p = pc(q), there exists some c(q) > 0 such that

π1Z2

((0, 1

2)∗←→ ∂J−n, nK2

)≤ e−cn . (5.1.1)

The geometric bottleneck described above then uses this property to bound the

conductance of the space.

The argument easily extends to (Z/nZ)× (Z/n′Z) where n′ = bαnc for some

aspect ratio α; moreover, it suffices to prove the theorem for the discrete-time

setting, since any O(n2) cost of moving to continuous time can be absorbed in the

exponential lower bound.

Proof of Theorem 2. In the context of this proof, denote by u! v the existence

of a sequence of sites uiki=1 such that ui is adjacent to ui+1 and with u1 = u and

136

D q

C q

Figure 5.1: Bottleneck for the Potts Glauber dynamics: the box Λ′ is equipartitionedinto three vertical and three horizontal strips. The green region is exposed to revealan outermost open circuit C q of color q (in this case, red). A boundary configurationσ ∈ ∂Siv,q contains a macroscopic dual-crossing in Dq like the dotted black pathextending from the pivotal vertex w to the inner boundary of its vertical strip.

uk = v, with σ(xi) = σ(xi+1) for all i. Define the bottleneck set

S =⋂

i=1,2,3

Siv,q ∩ Sih,q ,

where the constituent events are defined as follows for i = 1, 2, 3:

Siv,q = σ : ∃x such that (x, 0)! (x, n) in J (i−1)n3

, in3K× J0, nK, σ((x, 0)) = q ,

Sih,q = σ : ∃y such that (0, y)! (n, y) in J0, nK× J (i−1)n3

, in3K, σ((0, y)) = q .

We call each of these Potts connections of nontrivial homology on the torus Potts

loops, as the color clusters satisfying Siv,q (resp., Sih,q) will contain a path of homology

(0, 1) (resp., (1, 0)). Define the boundary subset ∂S = σ ∈ S : P (σ, Sc) > 0

where P (σ, ·) is the transition kernel for discrete-time Potts Glauber dynamics.

137

Notice that a configuration σ ∈ S is in ∂S if and only if there exists a vertex

v such that if the value of σv were changed to form σ′, then σ′ would not be in

S. The bound P (σ, σ′) ≤ 1 for all σ, σ′ implies that Q(S, Sc)/µpΛ(S) ≤ µpΛ(∂S | S),

where the superscript p indicate periodic boundary conditions on Λ = Λn. By

spin flip symmetry of the torus, and our definition of Potts loops (namely, the fact

that geometrically, Sih,q−1 ⊂ (Siv,q)c ⊂ Sc), we have µpΛ(Sc) ≥ 1

2. Therefore, the

conductance of the Markov chain satisfies

Φ? ≤Q(S, Sc)

µpΛ(S)µpΛ(Sc)≤ 1

2µpΛ(∂S | S) , (5.1.2)

and it suffices to show that the right-hand side is exponentially decaying in n.

We obtain an exponentially decaying upper bound on µpΛ(∂S | S) by examining

the pivotality of vertices to the event S. For a set S and a configuration σ ∈ S, we

call a vertex v pivotal to S if and only if there is a configuration σ′ /∈ S satisfying

σv 6= σ′v and σw = σ′w for every w 6= v, It is clear that the set

∂S =⋃w∈∂S

⋃i=1,2,3

⋃j∈v,h

σ ∈ S : w is pivotal to Sij,q .

Let us union bound over all vertices in Λ and the six different crossing events:

fix a vertex w ∈ Λ whose pivotality to—without loss of generality—Siv,q we examine.

Clearly, w must be in J (i−1)n3

, in3K× J0, nK. Choose a translate of Λ on the universal

cover by some (mn3, `n

3) to obtain Λ′, as follows:

1. vertically translate such that the horizontal third of Λ that contained w is

now the middle horizontal third of Λ′

2. If w is closer to the left than the right, of the vertical third of Λ containing it,

138

choose a horizontal translate of Λ so that vertical third becomes the left-most

vertical third of Λ′. Otherwise, horizontally translate so that vertical third

becomes the right-most vertical third of Λ′.

When considering instead pivotality to Sih,q, the roles of the two steps are reversed,

so that w will be translated to be in the middle vertical third of Λ′ and either

the top or bottom horizontal thirds. Notice that by periodicity of the boundary

conditions, and the invariance of the events Sij,q under translations by (mn3, `n

3)m,`∈Z,

that

µpΛ(w is pivotal to Sij,q | S

)= µpΛ′

(w is pivotal to Sij,q | S

).

In order to bound the probability of w being pivotal to S1v,q given S, let us reveal

the outermost Potts loops of color q in Λ′ as follows. First expose (sample from the

measure µpΛ′(σ∂Λ′ | S)) all the spin values on ∂Λ′ to reduce the torus to a rectangle

with fixed boundary conditions. Then given those exposed spins, iteratively expose,

starting from ∂Λ′, all spins ?-adjacent to (i.e., Euclidean distance at most√

2 away

from) exposed vertices whose spin value is not q. By construction, this revealing

process will end when we will have revealed the outermost q-colored paths, and

therefore a Potts circuit, C q, of spin value q, and nothing interior to it (such a

circuit exists since we are working conditionally on S).

Call Dq the set of all vertices in C q or “interior to it” (i.e., in the connected

subgraph of Λ′ \ C q containing the center of Λ′). By the domain Markov property,

the distribution on the remaining vertices in Dq is given by µσCq=qDq (· | S). Let us

now move to the FK representation of the model on Dq; the FK representation of

the has fully wired boundary conditions on C q, and therefore the event FσCq is

139

trivially satisfied. Refer to Figure 5.1 for a visualization.

In order for w to be pivotal to Siv,q it must be the case that w ∈ C q, so we

suppose that w ∈ C q. Also, we claim that, there must be a dual-crossing from one

of the interior dual-neighbors of w (an edge whose center is distance√

52

from v) to

one of n3× J0, nK or 2n

3× J0, nK in Λ′ (the choice depends on which translate Λ′

is chosen), and the dual-crossing must be contained completely within Dq. Indeed,

suppose there were no such dual-crossing. Then, by planarity, there would be a

primal FK connection in the same vertical third connecting the two neighbors of

w in C q. By the Edwards–Sokal coupling and the definition of Potts connections,

such an FK connection would translate to a new Potts connection of color q that

does not use the vertex w. The new vertical Potts crossing is still a vertical loop

because of the exposed horizontal crossings. in the circuit C q.

At the same time, by construction, the inner boundary of the vertical strip

containing w is at a graph distance at least n/6 from w. Since the event FσCq is

trivially satisfied, by monotonicity in boundary conditions, and the FKG inequality

(the FK representation of S is an increasing event and dual-crossings are decreasing),

we have uniformly over C q,

µpΛ′(w is pivotal to Siv,q | C q, S) ≤ πpΛ′( ⋃e:d(e,w)=

√5/2

e∗←→ n

3, 2n

3 × J0, nK | C q, S

)≤ 4π1

Z2

((0, 1

2)∗←→ ∂J−n

6, n

6K2).

In turn, by (5.1.1), this right-hand side is bounded above by e−cn for some c =

c(β, q) > 0 as soon as either q ≥ 1 and p > pc(q) or q > 4 and p = pc(q). Combining

140

this with (2.2.15) with (5.1.2), we see that

gapp ≤ µpΛ(∂S | S) ≤ n2e−cn ,

for every β, q such that the Potts model has distinct Z2-measures.

5.2 FK dynamics on (Z/nZ)2 at a discontinuous

phase transition

We now translate the proof in Section 5.1 to the FK Glauber dynamics; the idea

of the proof is essentially the same—we work directly with FK crossings rather than

loops of a fixed Potts color. It again suffices to prove the desired for the discrete-

time dynamics, as any polynomial in n cost of translating to continuous-time is

easily absorbed into the exponential slowdown.

Proof of Theorem 2. Using (5.1.1), we will establish a bottleneck set S for the

random cluster model on Λ with periodic boundary conditions. Define the bottleneck

event

S =⋂

i=1,2,3

Siv ∩ Sih

where the constituent events are defined as follows for i = 1, 2, 3:

Siv :=ω : ∃x such that (x, 0)←→ (x, n′) in J (i−1)n

3, in

3K× J0, n′K

,

Sih :=ω : ∃y such that (0, y)←→ (n, y) in J0, nK× J (i−1)n′

3, in′

3K.

The crossings in the above events will contain paths on the torus of homology

141

class (1, 0) and (0, 1), which we call horizontal and vertical loops, respectively. We

aim to get an exponentially decaying upper bound on πpΛ(∂S | S) (the superscript

p denotes periodic boundary conditions on Λ), where the boundary subset ∂S =

ω ∈ S : P (ω, Sc) > 0 is the event that there exists an edge e that is pivotal to S.

Specifically, we have

∂S =⋃

e∈E(Λ)

⋃i=1,2

⋃j∈v,h

ω ∈ S : ω − e /∈ Sij ,

in which configurations ω are identified with their edge-sets.

Via the symmetry of periodic FK boundary conditions, RSW-type estimates on

the torus (Z/nZ) × (Z/bαncZ) were proved in [5] for all q ≥ 1 at p = pc(q). By

[5, Theorem 5], therefore, there exists some ρ(α, q) > 0 such that πpΛ(C∗v(Jn3 ,2n3

K×

Jn′3, 2n′

3K) ≥ ρ, and since C∗v(Jn3 ,

2n3

K× Jn′3, 2n′

3K) ⊂ Sc, we get

Q(S, Sc)

πpΛ(S)πpΛ(Sc)≤ Q(S, Sc)

πpΛ(S)ρ≤ ρ−1πpΛ(∂S | S) .

Combining this with Eq. (2.2.15), it suffices to prove that there exist constants

c1 = c1(α, q) > 0 and c2 = c2(α, q) > 0 such that,

πpΛ(∂S | S) ≤ c1e−c2n . (5.2.1)

to obtain the desired bound on the inverse spectral gap. A union bound implies

πpΛ(∂S | S) ≤∑

e∈E(Λ)

∑i=1,2

∑j∈v,h

πpΛ(ω − e /∈ S | ω ∈ S) .

Without loss of generality, examine the probability that e is pivotal to Siv. The

142

D

C

Figure 5.2: The lower bound construction for the FK Glauber dynamics: the boxΛ′ is equipartitioned into three vertical and three horizontal strips. The purplecrossings form an outermost open circuit C . A boundary configuration ω ∈ ∂Scontains a macroscopic dual-crossing like the red path extending from the pivotaledge e to the boundary of the vertical strip.

other cases can be treated analogously. Fix an edge e in J0, n3K× J0, n′K and consider

its horizontal coordinate (in Fig. 5.2, e is the edge of intersection of the purple and

red paths).

Now move to a translate by (mn3, `n

′

3)m,`∈Z of Λ on the universal cover of the

torus, which we call Λ′ = Λ′n,n′ . Choose a translate such that:

1. The horizontal third of Λ that contained e, is now the middle horizontal third

of Λ′.

2. If e was closer to the left of its vertical third than the right, that strip is the

left vertical third of Λ′. Otherwise, that strip is the right vertical third of Λ′.

As before, by periodicity of boundary conditions and invariance of Sij under

translations by (mn3, `n

′

3)m,`∈Z, it suffices to bound the probability that e is pivotal

143

to the event S in Λ′. Begin by exposing all the edges E(∂Λ′) so as to fix a boundary

condition and move from a torus to a rectangle with some fixed boundary condition.

Then expose the outermost circuit C in Λ′ as follows (similarly to steps (1)–(4)).

First expose the left-most vertical crossing of Λ′ by revealing the dual component

of ∂wΛ′: by construction, its adjacent primal edges will form the left-most vertical

crossing of Λ′ and we will not have revealed any edges to its right. Repeating this

procedure on the n,e, s sides reveals an outermost circuit C without exposing any

of its interior edges (see Fig. 5.2 where the shaded region consists of the edges we

reveal through this process).

If e is not an open edge in one of the vertical crossings we have exposed, then

certainly e is not pivotal to the event S. So suppose e is an open edge in a vertical

loop and—by construction—also an open edge in C . Denote by D the subgraph of

Λ′ whose edge values have not yet been exposed, i.e., all edges interior to C .

It is a necessary condition for e to be pivotal to S that there exists a dual path

from the dual-vertex in D incident to the dual-edge e∗ to the vertical boundary of

the vertical third containing e in Λ′, i.e., to n3, 2n

3 × J0, n′K. If there is no such

dual-path, then there is a different primal vertical crossing of that strip which does

not contain e, and that crossing—because of the exposed horizontal loops—is itself

a loop of homology class (0, 1) and satisfies S1v .

But observe that n3, 2n

3× J0, n′K is a graph distance at least n/6 from e. Thus,

by the Domain Markov property and FKG inequality, for every possible circuit C

revealed,

πpΛ′(ωD ∈ · | S) = π1C∪D(· | S) π1

C∪D π1Z2(ωD ∈ ·) .

By (5.1.1), for every q > 4, if p = pc(q), there exists an absolute c(q) > 0 such

144

that, for any fixed e,

πpΛ(ω − e /∈ Siv | ω ∈ S) ≤ e−cn .

If e were pivotal to Sih the analogous claim would hold with probability less than

e−cn′/6. Summing over all six crossing events and summing over all edges e we

conclude that πpΛ(∂S | S) ≤ 12αn2e−cαn, implying Eq. (5.2.1) as desired.

5.3 Potts Glauber dynamics with free boundary

at β > βc(q)

We conclude the section with a short proof that the mixing time of the Potts

Glauber dynamics on Λn with free boundary conditions is exponentially slow at

β > βc(q). (N.b. unlike the periodic boundary conditions, this slow mixing should

not extend to the critical point β = βc(q), even for q > 4, where the mixing time

should be polynomial: see Problem 1.5.5.) The proof here uses a similar bottleneck

set and revealing procedure to that in Section 5.1 based on vertical and horizontal

crossings of Λn but is substantially simplified by the uniqueness of the FK measure

when p > pc(q).

Theorem 5.3.1. Let q ≥ 2 and β > βc(q). There exist c(β, q), c′(β, q) > 0 such

that Potts Glauber dynamics on Λn with free boundary conditions satisfies

c ≤ 1

nlog gap−1

p ≤ c′ .

Proof. Recall the notation u! v indicating a sequence of monochromatic edges

145

(the endpoints of the edges have the same state) connecting vertices u to v, and

define the bottleneck set S = Sv,q ∩ Sh,q where

Sv,q = σ : ∃u ∈ ∂sΛn, v ∈ ∂nΛn such that u! v, σ(u) = q ,

Sh,q = σ : ∃u ∈ ∂eΛn, v ∈ ∂wΛn such that u! v, σ(u) = q .

Recall that the boundary subset ∂S is the set σ ∈ S : P (σ, Sc) > 0 (where

P (σ, ·) is the transition kernel for discrete-time Potts Glauber dynamics. Then

a configuration σ ∈ S is in ∂S if and only if there exists a pivotal vertex w, i.e.,

if σw were changed, the resulting configuration would not be in S. By the fact

that Sh,q−1 ⊂ Scv,q ⊂ Sc, we have µΛ(Sc) ≥ 12. As in (5.1.2), it therefore suffices to

obtain an exponentially decaying upper bound on µΛ(∂S | S).

As before, let us union bound over the n2 possible vertices w ∈ Λn and whether

they are pivotal to Sv,q or Sh,q. Let us fix any w ∈ Λn and consider the event w

is pivotal to Sv,q; the bound on pivotality of w ∈ Λn to Sh,q follows analogously

(consider the rotation by π2

of Λn). Suppose w is in the right-half of Λn, i.e.,

w ∈ Jdn2e, nK × J0, nK (resp., left-half w ∈ J0, bn

2cK × J0, nK) reveal the right-most

(resp., left-most) vertical Potts crossing of color q in Λ as in the proof of Theorem 2.

Namely, in the case that we are revealing the right-most vertical crossing, iteratively

expose, starting from ∂eΛ, the states of vertices ?-adjacent to exposed vertices

whose state is not q. This revealing process will end when we have revealed the

right-most q-colored vertical crossing (such a crossing exists since we are doing

this revealing under the conditional measure µΛ(· | S)). Call this vertical q-colored

crossing C qv and call Dq

v the set of all vertices in C qv along with all those whose

value have not yet been exposed (those to the left of C qv ). By the domain Markov

146

property, the distribution on the remaining vertices in Dq is given by µσCq

v=q

Dqv

whose

FK representation is wired on C qv and free on ∂Λn ∩ ∂Dq

v . Since this FK boundary

condition has one wired boundary component (corresponding to C qv ), the event

FσCqv

is trivially satisfied.

In order for w to have been pivotal to Sv,q it must be the case that w ∈ C qv and

moreover, that one of the two dual-vertices adjacent to w and interior to Dqv are

dual-connected to ∂wΛn. Otherwise, by planar duality, ∂nΛn would be connected to

∂sΛn by an open path that intersects C qv and does not use w. By monotonicity

in boundary conditions and the FKG property, we therefore have

µΛn(w is pivotal to Sv,q | S) ≤ π0Λn

( ⋃e:d(e,w)=

√5/2

e∗←→ ∂wΛn

)

which is at most 4ne−cd(∂wΛ,w) ≤ 4ne−cn/2 for some c(p, q) > 0 by the exponential

decay of dual-connectivities at p > pc(q) (the dual of (2.1.1)).

Summing over all w ∈ Λn and the vertical and horizontal crossing events, we

deduce that

µΛn(∂S | S) ≤ µΛn(∃w : w is pivotal to S | S) ≤ e−cn

for some c(p, q) > 0, concluding the proof.

147

Chapter 6

FK boundary conditions in

uniqueness regions

In Chapters 4–5, we analyzed the mixing times of Potts and FK Glauber

dynamics on the torus (Z/nZ)2. As we saw, the bounds at a continuous phase

transition point—where there is a unique infinite-volume measure—in fact applied

much more broadly, to boxes with arbitrary boundary conditions in the Potts case,

and “typical” boundary conditions in the FK case. When there is no uniqueness

of infinite-volume measure, mixing times are very sensitive to choice of boundary

condition, as is well-studied in the low-temperature Ising model, and explored

further for the Potts and FK models in Chapters 7–8. Before that, let us first

investigate the effect that FK boundary conditions can have on mixing times in

the off-critical uniqueness regimes of p 6= pc(q).

For the Ising and Potts models, at high-temperatures, where correlations die

out exponentially fast even adjacent to the boundary conditions—a property known

as strong spatial mixing (SSM)—it is well known that the mixing time of Ising and

148

Potts Glauber dynamics are fast, independently of the boundary conditions [2, 69].

In the FK model, the analogous strong spatial mixing property does not hold near

the boundary even for “typical” FK boundary conditions. As such, the work of [10]

proving fast mixing for the FK dynamics when p 6= pc(q) only held on the select

FK boundary conditions for which SSM does hold, namely periodic, wired, and

free. In this chapter, we analyze the stability of those mixing time bounds with

respect to changes of FK boundary condition, proving Theorems 5–6.

This chapter is organized as follows. In Section 6.0.1, we introduce our general

framework to deduce mixing time estimates on Λn from spatial and local mixing

properties. The key ingredient for us is a mixing time bound for thin rectangles

(Theorem 6.1.1) with arbitrary realizable boundary conditions, presented in Sec-

tion 6.1. This proof uses a novel block-dynamics approach in the presence of FK

boundary conditions, recursing not over rectangles but rather on collections of

rectangles dictated by the boundary conditions at each scale (see Section 6.1.1 for

a proof sketch). We are then in position to complete the proof of Theorem 5 in

Section 6.2, showing that when p 6= pc(q), uniformly over all realizable boundary

conditions ξn, the mixing time on Λξnn is at most polynomial in n. This is boosted

to nearly-optimal mixing time for typical boundaries (Theorem 6) in Section 6.3.

We also complement these with two results of a different flavor. As mentioned in

Remark 2.2.11, the traditional canonical paths bound on thin rectangles (exponential

in the shorter side length) does not apply in the presence of FK boundary conditions

as they play the role of distorting the underlying graph; we are nonetheless able to

prove a bound of a similar flavor under realizable boundary conditions in Section 6.4.

Finally, we prove the slow mixing result for worst-case (non-realizable) boundary

conditions in (Theorem 7) in Section 6.5.

149

For ease of exposition, we will prove everything in this chapter for the discrete-

time FK Glauber dynamics—the inverse spectral gap and mixing time of the

continuous-time dynamics are seen to be an order n2 faster by standard comparison

inequalities.

6.0.1 Mixing time upper bounds: a general framework

In this section we introduce a general framework for bounding the mixing time

of the FK-dynamics on Λn = (Λn, E(Λn)) by its mixing times on certain subsets.

In [10] it was shown that a strong form of spatial mixing (encoding exponential

decay of correlations uniformly over subsets of Λn) implies optimal mixing of the

FK-dynamics. However, this notion, known as strong spatial mixing (SSM) and

described in Remark 6.0.2, does not hold for most boundary conditions for which

fast mixing of the FK-dynamics is still expected. To circumvent this, we introduce

a weaker notion, which we call moderate spatial mixing (MSM).

Notation. We introduce some notation first. For a set R ⊆ Λn, let E(R) ⊆ En be

the set of edges of E(Λn) with both endpoints in R. We will denote by Rc the vertex

set Λn \R and by Ec(R) the edge-complement of R; i.e., Ec(R) := E(Λn) \ E(R).

For a configuration ω : E(Λn)→ 0, 1, we will use ω(R), or alternatively ω(E(R)),

for the configuration of ω on E(R). With a slight abuse of notation, for an edge

set F ⊆ E(Λn), we use F = ω for the event that the configuration on F is given

by ω; when ω is the all free or the all wired configuration, we simply use F = 0

and F = 1, respectively.

Definition 6.0.1. Let ξ be a boundary condition for Λn = (Λn, E(Λn)) and let

B = B1, B2, . . . , Bk be a collection of subsets of Λn. We say that moderate spatial

150

mixing (MSM) holds on Λn for ξ, B and δ > 0 if for all e ∈ E(Λn), there exists

Bj ∈ B such that

∣∣∣πξΛn,p,q( e = 1 | Ec(Bj) = 1 )− πξΛn,p,q( e = 1 | Ec(Bj) = 0 )∣∣∣ ≤ δ . (6.0.1)

In words, MSM holds for B if for every edge e ∈ E(Λn) we can find Bj such that

e ∈ E(Bj) and the “influence” of the configuration on Ec(Bj) on the state of e is

bounded by δ.

Remark 6.0.2. SSM as defined in [10] holds when MSM holds for a specific

sequence of collections of subsets: if Br is the set of subsets containing all the

square boxes of side length 2r centered at each e ∈ E(Λn) (intersected with E(Λn)),

then SSM holds if MSM holds for Br for every r ≥ 1 with δ = exp(−Ω(r)).

MSM does not capture the fast mixing of the FK-dynamics the way SSM

does. Namely, it is easy to find collections of subsets for which MSM holds for all

boundary conditions, including those boundary conditions for which we later prove

slow mixing; see Theorem 7. However, if, for a collection B = B1, B2, . . . , Bk,

we also bound the mixing time of the FK-dynamics on every Bj, we can deduce a

mixing time bound for the FK-dynamics on Λn. Let tmix(Bτ ) denote the mixing

time of the FK-dynamics on the subset B ⊆ Λn with boundary condition τ . (Recall

that τ corresponds to a partition of ∂B and that ∂B consists of those vertices in B

that are adjacent to vertices in Z2 \B.)

Definition 6.0.3. Let ξ be a boundary condition for Λn = (Λn, E(Λn)) and let

B = B1, B2, . . . , Bk with Bj ⊂ Λn. We say that local mixing (LM) holds for B

151

and T > 0, if

tmix(B

(1,ξ)j

)≤ T and tmix

(B

(0,ξ)j

)≤ T for all j = 1, ..., k

where (1, ξ) (resp., (0, ξ)) denotes the boundary condition on Bj induced by the

event Ec(Bj) = 1 (resp. Ec(Bj) = 0) and the boundary condition ξ.

Remark 6.0.4. Observe that when Bj ∩ ∂Λn = ∅, (1, ξ) and (0, ξ) are simply the

wired and free boundary condition on Bj, respectively. When Bj ∩ ∂Λn 6= ∅, the

connectivities from ξ could also induce some connections in (1, ξ) and (0, ξ).

Our next theorem, roughly speaking, establishes the following implication:

MSM + LM =⇒ upper bound for mixing time of FK-dynamics,

with the quality of the bound depending on the T for which LM holds. A similar

(and inspiring) implication for the Glauber dynamics of the Ising model in graphs

of bounded degree was established by Mossel and Sly in [73]; there, the notion of

MSM is replaced by a form of spatial mixing which is stronger than SSM.

Theorem 6.0.5. Let ξ be a boundary condition on Λn = (Λn, E(Λn)) and let

B = B1, B2, . . . , Bk with Bj ⊂ Λn for all j = 1, . . . , k. If for ξ and B, moderate

spatial mixing holds for some δ ≤ 1/(12|E(Λn)|) and local mixing holds for some

T > 0, then tmix(Λξn) = O(Tn2 log n).

Proof. Consider two copies Xt, Yt of the FK-dynamics. We couple the evo-

lution of Xt and Yt by using the same random e ∈ E(Λn) and the same

uniform random number r ∈ [0, 1] to decide whether to add or remove e from the

configurations. This is a standard coupling of the steps of the FK-dynamics (see,

152

e.g., [10, 47]); we call it the identity coupling. It is straightforward to verify that,

when q ≥ 1, the identity coupling is a monotone coupling, in the sense that if

Xt ⊆ Yt then Xt+1 ⊆ Yt+1 with probability 1.

We bound the coupling time Tcoup of the identity coupling for the FK-dynamics.

The result then follows from the fact that tmix ≤ Tcoup. Since the identity coupling

can be extended to a simultaneous coupling of all configurations that preserves

the partial order ⊆, the coupling time starting from any pair of configurations is

bounded by the coupling time for initial configurations Y0 = ∅ and X0 = E(Λn).

We prove that there exists T = O(Tn2 log n) such that the identity coupling P

satisfies

P[XT (e) 6= YT (e)] ≤ 1

4|E(Λn)|

for every e ∈ E(Λn). A union bound over e ∈ E(Λn) then implies that Tcoup ≤ T .

Fix any edge e ∈ E(Λn) and let Bj ∈ B be a subset for which (6.0.1) is

satisfied (such a subset exists since moderate spatial mixing holds). To bound

P[XT (e) 6= YT (e) ], we introduce two additional instances Z+t , Z−t of the FK-

dynamics. These two copies update only edges with both endpoints in Bj and

reject all other updates. We set Z+0 = E(Λn) and Z−0 = ∅. The four Markov chains

Xt, Yt, Z+t and Z−t are coupled with the identity coupling, with updates

outside Bj ignored by both Z+t and Z−t . The monotonicity of this coupling,

along with a triangle inequality, implies that for all t ≥ 0,

P[Xt(e) 6= Yt(e) ] ≤∣∣∣P[Z+

t (e) = 1 ]− πξΛn( e = 1 |Ec(Bj) = 1 )∣∣∣ (6.0.2)

+∣∣∣πξΛn( e = 1 |Ec(Bj) = 1 )− πξΛn( e = 1 |Ec(Bj) = 0 )

∣∣∣+∣∣∣πξΛn( e = 1 |Ec(Bj) = 0 )− P[Z−t (e) = 1 ]

∣∣∣ . (6.0.3)

153

Observe that the restrictions of the chains Z+t and Z−t to Bj are lazy ver-

sions of FK-dynamics on Bj with stationary measures πξΛn( · |Ec(Bj) = 1 ) and

πξΛn( · |Ec(Bj) = 0) , respectively. The laziness comes from the fact that they only

accept updates that are in Bj , and by the local mixing assumption, once T updates

have been done in Bj, the chains Z+t and Z−t will be mixed.

Now, after T = CT |E(Λn)| log2d24|E(Λn)|e steps, the expected number of

updates in Bj is

CT |E(Λn)| log2d24|E(Λn)|e |E(Bj)||E(Λn)|

≥ CT log2d24|E(Λn)|e .

Let AT be the event that the number of updates in Bj after T steps is at least

T log2d24|E(Λn)|e. A Chernoff bound then implies that, for a large enough constant

C > 0,

Pr[AcT

] ≤ 1

24|E(Λn)|.

Therefore,

∣∣∣Pr[Z+

T(e) = 1 | AT ]− Pr[Z+

T(e) = 1]

∣∣∣ ≤ Pr[AcT

] ≤ 1

24|E(Λn)|. (6.0.4)

By the local mixing property, and the fact that tmix(ε) ≤ dlog2 ε−1e · tmix for

any positive ε < 1/2, we have

∣∣∣Pr[Z+

T(e) = 1 | AT ]− πξΛn( e = 1 |Ec(Bj) = 1 )

∣∣∣ ≤ 1

24|E(Λn)|. (6.0.5)

It then follows from (6.0.4), (6.0.5) and the triangle inequality that when t = T

the right-hand side in (6.0.2) is at most 112|E(Λn)| . The same bound can be deduced

for (6.0.3) in a similar manner.

154

Finally, since moderate spatial mixing holds for some δ < 112|E(Λn)| , we have

∣∣∣πξΛn( e = 1 |Ec(Bj) = 1 )− πξΛn( e = 1 |Ec(Bj) = 0 )∣∣∣ ≤ 1

12|E(Λn)|;

see Definition 6.0.1. Putting these together we see that P[XT (e) 6= YT (e) ] ≤ 14|E(Λn)|

as desired.

6.1 Fast mixing on thin rectangles

The main difficulty in proving Theorem 5 using the general framework from

Section 6.0.1 is obtaining mixing time estimates on thin rectangles of dimension

Θ(n)×Θ(log n) with realizable boundary conditions. To motivate this, we notice

that since p < pc(q), the influence of the boundary is lost, with high probability,

at a distance Θ(log n). Thus the main difficulty will be to bound the mixing time

of the FK-dynamics in the annulus of width Θ(log n) with realizable boundary

conditions on the outside. As such, the key ingredient in the proof of Theorem 5

will be the following mixing time bound on thin rectangles.

As before, for an n × l rectangle Λn,l = J0, nK × J0, lK, we use ∂nΛn,l, ∂eΛn,l,

∂sΛn,l and ∂wΛn,l for its north, east, south and west boundaries respectively.

Theorem 6.1.1. Consider Λn,l = (Λn,l, E(Λn,l)) for l ≤ n with an arbitrary

realizable boundary condition ξ that is either free or wired on ∂eΛn,l∪∂wΛn,l∪∂sΛn,l.

Then, for every q > 1 and p 6= pc(q), the mixing time of the FK-dynamics on Λn,l

is at most exp(O(l + log n)).

Observe that when l = O(log n), this implies the mixing time is nO(1), which will

be the setting of interest in our proofs. Moreover, we note that it suffices for us to

155

Bw Be

(a)

R1R2 R2

(b)

Figure 6.1: (a) A boundary condition for which no configuration in Bw∩Be isolatesBw \Be from Be \Bw. (b) A boundary condition ξ where every pair of overlappingrectangles (as in (6.1.1)) must interact through ξ; however, the two groups ofrectangles R1, R2 do not interact through ξ.

prove Theorem 6.1.1 for the set of realizable boundary conditions ξ that are free on

∂eΛn,l ∪ ∂wΛn,l ∪ ∂sΛn,l and all p 6= pc(q), as the set of boundary conditions dual

to these are exactly the set of realizable boundary conditions that are wired on

∂eΛn,l ∪ ∂wΛn,l ∪ ∂sΛn,l; see Remark 2.2.1.

In Section 6.1.1, we give a overview of the main ideas in the proof of this theorem.

In Sections 6.1.2–6.1.4, we introduce some crucial notions regarding groups of

rectangular subsets of Λn,l and their relations with the boundary conditions ξ.

Sections 6.1.5–6.1.6 bound the mixing time of the block dynamics with respect to

a suitably-chosen set of subsets of Λn,l. The recursive proof of Theorem 6.1.1 is

then completed in Section 6.1.7.

6.1.1 Outline of proof of Theorem 6.1.1

We first mention some obstructions that boundary conditions present to proving

Theorem 6.1.1 using approaches that are common in analogous problems for spin

systems. A traditional approach to proving mixing time bounds for thin rectangles

is the canonical paths method ([53, 66, 67, 86]), which gives an upper bound that

is exponential in the shorter side length (see Lemma 2.2.10); however, boundary

conditions can significantly distort the augmented graph with external wirings,

156

preventing this approach from succeeding. (Though this is not the approach we take,

we are able to obtain a weaker canonical paths bound using a tree-like structure of

realizable boundary conditions that holds for all p, q in Section 6.4).

A sharper approach would be to use an inductive scheme (e.g., as in [18, 64, 67]

as well as our Chapter 4), whereby, we bound the mixing time of the FK Glauber

dynamics on Λn,l by the mixing times in smaller rectangular blocks, e.g.,

Bw = J0, 23nK× J0, lK and Be = J1

3n, nK× J0, lK . (6.1.1)

This method requires bounding by the mixing time of the so-called block dynamics

as defined for the Potts Glauber dynamics in Section 2.2.2 and for the FK Glauber

dynamics in Section 6.1.6.

The mixing time of the FK Glauber dynamics on Λn,l is bounded by the mixing

time of the block dynamics times the worst mixing time of the FK dynamics in any

Bi with worst-case configuration on Ec(Bi); see Theorem 6.1.18. With the choice

of blocks in (6.1.1), applying this recursively, one would bound the mixing time

of the FK dynamics on Λn,l by the mixing time of the block dynamics raised to a

Θ(log n) power. Hence, establishing Theorem 6.1.1 would require an O(1) upper

bound on the mixing time of the block dynamics.

As in Chapter 4, the mixing time and spectral gap of the block dynamics is

typically bounded by showing that after the first block update in either Bw or Be,

the configuration in Bw ∩Be will be such that it disconnects the influence of the

configuration on Bw \Be from Be \Bw with probability Ω(1); this then allows a

standard coupling argument to be used to bound the mixing time. In the presence

of long-range boundary connections, however, it could be that no configuration on

157

Bw ∩Be would disconnect the two sides from one another and facilitate coupling;

see Figure 6.1(a) for such an example. As such, our choices of blocks will depend

on the boundary conditions, and be chosen to allow for the block dynamics to

couple in O(1) time, while ensuring that the blocks are still at most a fraction of

the size of the original rectangle, so that after O(log n) recursive steps we arrive at

a sufficiently small base scale.

In particular, we will show that for every realizable boundary condition ξ on

Λn,l, there exists a choice of two blocks, whose widths are at most 45n, such that

they are sufficiently isolated from one another in ξ. As Figure 6.1(b) demonstrates,

there are realizable boundary conditions that would force these blocks to not be

single rectangles, as in (6.1.1), but rather collections of rectangular subsets of Λn,l.

Thus, our recursive argument will proceed instead on groups of rectangles,

R =⋃

Ri, where Ri = Jai, biK× J0, lK ⊂ Λn,l

are disjoint, with boundary conditions induced by ξ and the configuration of the

chain on E(Λn,l) \ E(R).

We formally define groups of rectangles (Definition 6.1.6), and their boundary

conditions in Section 6.1.3. We consider next the notion of compatibility of a

group of rectangles R ⊂ Λn,l with the boundary condition ξ. Roughly speaking,

we say a group of rectangles R is compatible with ξ if ξ limits the boundary

interactions between the rectangles of R for every possible configuration on Λn,l \R;

see Definition 6.1.8. In Section 6.1.5, we provide an algorithm (see Lemma 6.1.10)

that, for a group of rectangles R compatible with ξ, finds two suitable subsets

for the block dynamics: these subsets will each be group of rectangles compatible

158

with ξ, of width between 1/5 and 4/5 of the width of R. This will allow us to

induct on groups of rectangles compatible with ξ, while ensuring that number of

recursive steps is O(log n). Specifically, our splitting algorithm will find interior

and exterior subsets Aint and Aext with no boundary connections between the two;

see Figure 6.3(a). These will be the cores of the two blocks for R, but in order to

bound the coupling time of the block dynamics, we want the two blocks to overlap,

and therefore we enlarge Aint and Aext by m = Θ(log l) to form the blocks Rint

and Rext for the block dynamics; see Figure 6.3(b).

Finally, in Section 6.1.6, we bound the coupling time of this block dynamics by

some sufficiently large constant to conclude the proof. This follows by leveraging

the fact that p < pc (p > pc) to condition on the existence of certain disconnecting

dual (primal) paths in Rint ∩Rext that have Ω(1) probability.

We emphasize that in order to push through this recursion, is will be crucial

that our notion of compatibility with ξ is strong enough to yield a uniform bound

on this block dynamics coupling time, while being broad enough that the splitting

algorithm always succeeds in finding sub-blocks that are themselves groups of

rectangles compatible with ξ.

6.1.2 Disconnecting intervals

In this section, we introduce the notion of disconnecting intervals, one of the

building blocks of our recursive proof of polynomial mixing. Recall that we use

∂nΛn,l, ∂eΛn,l, ∂sΛn,l and ∂wΛn,l for the north, east, south and west boundaries of

the rectangle Λn,l, respectively.

Definition 6.1.2. For a realizable boundary condition ξ on Λn,l that is free on

∂eΛn,l ∪ ∂sΛn,l ∪ ∂wΛn,l, an interval Ja, bK ⊂ J0, nK is called disconnecting of

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a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10

Figure 6.2: The rectangle Λn,l with a boundary condition ξ inducing disconnectingintervals. For example, Ja1, a4K, Ja3, a4K and Ja7, a10K are disconnecting intervalsof free-type; Ja1, a2K, Ja0, a6K, Ja7, a8K and Ja9, a10K are of free-wired-type; Ja0, a5Kand Ja5, a6K are of wired-type.

1. free-type: if there are no boundary connections in ξ between ∂Λn,l∩(Ja, bK×l)

and ∂Λn,l ∩ (Ja, bKc × J0, lK)

2. wired-type: if there is a boundary component in ξ that contains the vertices

(a, l) and (b, l), as well as possibly other vertices in ∂Λn,l.

Observe that an interval can be both of free-type and of wired-type if (a, l) and

(b, l) are connected through ξ but are not connected to any boundary vertex in

Ja, bKc × J0, lK; in this case, we may refer to the interval as being of free-wired-type;

see Figure 6.2 for several examples.

The following properties concerning the union and intersection of disconnecting

intervals will be crucial to our proofs.

Lemma 6.1.3. Let ξ be a realizable boundary condition on Λn,l that is free on

∂sΛn,l ∪ ∂eΛn,l ∪ ∂wΛn,l and let a < b < c. If both Ja, bK and Jb, cK are disconnecting

intervals of wired-type, then so is Ja, cK. If both Ja, bK and Jb+1, cK are disconnecting

intervals of free-type, then so is Ja, cK.

Proof. If Ja, bK and Jb, cK are disconnecting intervals of wired-type, then by definition

the vertices (a, l), (b, l) and (c, l) are all in the same component of ξ; hence Ja, cK

is a disconnecting interval of wired-type. If Ja, bK and Jb+ 1, cK are disconnecting

intervals of free-type, then by definition there are no connections in ξ between Ja, bK

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and Ja, bKc ⊃ Ja, cKc or between Jb + 1, cK and Jb + 1, cKc ⊃ Ja, cKc. Consequently,

there are not connections in ξ between Ja, cK = Ja, bK ∪ Jb + 1, cK and Ja, cKc, and

Ja, cK is thus a disconnecting interval of free-type.

Lemma 6.1.4. Let ξ be a realizable boundary condition on Λn,l that is free on

∂sΛn,l ∪ ∂eΛn,l ∪ ∂wΛn,l. Suppose there exist a < b ≤ c < d such that Ja, cK and

Jb, dK are disconnecting intervals. Then either both Ja, cK and Jb, dK are of free-type

or both are of wired-type.

Proof. Suppose by way of contradiction and without loss of generality that Ja, cK is

only of free-type (i.e., free-type but not free-wired-type) and Jb, dK is of wired-type.

By definition, there exists a component of ξ that contains both (b, l) and (d, l).

Since a < b ≤ c < d, there is therefore a connection in ξ between Ja, cK× l and

Ja, cKc × l. Hence, Ja, cK cannot be a disconnecting interval of free-type yielding

the desired contradiction.

Lemma 6.1.5. Let ξ be a realizable boundary condition on Λn,l that is free on

∂sΛn,l ∪ ∂eΛn,l ∪ ∂wΛn,l. Suppose there exist a < b < c < d such that Ja, cK and

Jb, dK are disconnecting intervals.

1. If Ja, cK and Jb, dK are both of wired-type, then Ja, bK, Jb, cK, Jc, dK and Ja, dK

are all disconnecting intervals of wired-type.

2. If Ja, cK and Jb, dK are both of free-type, then Ja, b− 1K, Jb, cK, Jc+ 1, dK and

Ja, dK are all disconnecting intervals of free-type.

Proof. For the first part, suppose that both Ja, cK and Jb, dK are disconnecting

intervals of wired-type. By definition, the vertices (a, l) and (c, l) are in the same

component of ξ, as are (b, l) and (d, l). By the planarity of realizable boundary

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conditions, it must be the case that these two components of ξ are indeed the same,

and therefore (a, l), (b, l), (c, l), (d, l) are all in the same boundary component of ξ.

As such, Ja, bK, Jb, cK, Jc, dK and Ja, dK are all disconnecting intervals of wired-type.

For the second part, suppose first by way of contradiction that Ja, b− 1K is not a

disconnecting interval of free-type. Then there must be a boundary component in ξ

with vertices in Ja, b− 1K×l and Ja, b− 1Kc×l. If it is a component containing

a vertex in Ja, cKc × l, it would violate the fact that Ja, cK is disconnecting of

free-type, while if it is a component containing a vertex in Jb, cK × l, it would

violate that Jb, dK is disconnecting of free-type as Ja, b− 1K ⊂ Jb, dKc. By analogous

reasoning Jc+ 1, dK, Jb, cK and Ja, dK are all disconnecting intervals of free-type.

6.1.3 Groups of rectangles

In this section, we define groups of rectangles and their boundary conditions,

which constitute the other building blocks of our recursive proof of polynomial

mixing for the FK-dynamics on thin rectangles. As in the previous section, we

consider an n× l rectangle Λn,l = J0, nK×J0, lK with a realizable boundary condition

ξ that is free on ∂eΛn,l ∪ ∂sΛn,l ∪ ∂wΛn,l.

Henceforth, we take

m = m(l) = C? log l ,

where C? is a large constant such that C? > c−1 which we choose later, with c being

the constant from (2.1.1).

A rectangular subset R ⊂ Λn,l is a rectangle of the form Ja, bK× J0, lK for some

0 ≤ a < b ≤ n. For such a rectangular subset, we denote by W (R) its width;

i.e., W (R) = b − a. For the union of distinct and disjoint rectangular subsets

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R =⋃N(R)i=1 Ri, where Ri = Jai, biK× J0, lK and a1 < b1 < a2 < · · · < aN(R) < bN(R),

its width is defined by W (R) =∑N(R)

i=1 W (Ri) and its boundary is given by

∂R =⋃N(R)i=1 ∂Ri. Moreover, we let ∂nR =

⋃N(R)i=1 ∂nRi and similarly define ∂sR;

on the other hand, ∂eR will be the eastern boundary of the right-most rectangle

Ri and ∂wR, the western boundary of the left-most.

Definition 6.1.6. A group of rectangles R =⋃N(R)i=1 Ri is the union ofN(R) disjoint

rectangular subsets Ri of Λn,l such that W (Ri) ≥ 2m for every i = 1, ..., N(R).

Remark 6.1.7. The requirement that W (Ri) ≥ 2m for every i, which may seem

arbitrary at the moment, is because in our recursive argument, we want our groups

of rectangles R to have interiors that are not influenced by the configuration on

E(Λn,l) \ E(R). When a group of rectangles has a thin constituent rectangle Ri,

the influence of the outside configuration can permeate through all of Ri.

We will be considering blocks dynamics with blocks consisting of groups of

rectangles. If we want to update the configuration on a group of rectangles R ⊂ Λn,l,

the boundary condition induced on ∂R will consist of the boundary condition on

Λn,l which will be fixed to be ξ, together with a fixed random-cluster configuration

ωRc on Ec(R) = E(Λn,l) \ E(R). Hence, our boundary conditions on R will be of

this form, and we denote them by the pair ζ = (ξ, ωRc).

6.1.4 Compatible boundary conditions

We now define the notion of compatibility of groups of rectangles with boundary

conditions ξ. This is crucial in our inductive argument; our algorithm for finding

suitable blocks for the block dynamics will guarantee that if the starting group

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rectangles is compatible w.r.t. to a given boundary condition, so will each of the

blocks, enabling an inductive procedure.

Let ξ be a realizable boundary condition on ∂Λn,l that is free on ∂sΛn,l∪∂eΛn,l∪

∂wΛn,l, and free in all vertices in ∂nΛn,l at distance at most m from ∂eΛn,l ∪ ∂wΛn,l

(i.e., they appear as singletons in the corresponding boundary partition). This latter

requirement for vertices near the corners of Λn,l allows us to always choose blocks

in our recursion whose boundaries are at least distance m from ∂eΛn,l ∪ ∂wΛn,l; this

simplifies our analysis.

The following will be the distinguishing property of our choice of blocks for the

block dynamics.

Definition 6.1.8. Let R =⋃N(R)i=1 Ri be a group of rectangles with Ri = Jai, biK×

J0, lK and a1 < b1 < a2 < . . . < aN(R) < bN(R). We say R is compatible with ξ, if

1. Between every two consecutive rectangles Ri = Jai, biK × J0, lK and Ri+1 =

Jai+1, bi+1K× J0, lK the interval Jbi −m, ai+1 +mK is a disconnecting interval;

2. The interval Ja1 +m, bN(R) −mK is also a disconnecting interval.

Remark 6.1.9. It is clear from the definition that Λn,l is compatible with ξ: the

first condition is vacuous, while the second is satisfied since all vertices a distance

at most m from ∂eΛn,l ∪ ∂wΛn,l are free. Observe also that bi − ai ≥ 2m for every i

and so bN(R) −m ≥ a1 +m; see Definition 6.1.6.

6.1.5 Defining the blocks for the block dynamics

Now that we have introduced disconnecting intervals, group of rectangles and

the notion of compatibility, we describe our algorithm for picking two blocks

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Rint,Rext for the block dynamics, based on the boundary condition ξ. Recall

the definitions of width W (·) of a rectangular subset and width of a collection of

rectangles W (R) =∑N(R)

i=1 W (Ri). It will also be convenient to have the following

notation ∂‖R =⋃N(R)i=1 ∂wRi ∪ ∂eRi for the vertical sides of the group of rectangles

R =⋃N(R)i=1 Ri. The following lemma provides the basis for our splitting algorithm.

Lemma 6.1.10. Let ξ be a realizable boundary condition on ∂Λn,l that is free

on ∂sΛn,l ∪ ∂eΛn,l ∪ ∂wΛn,l and free in all vertices in ∂nΛn,l at distance at most

m from ∂eΛn,l ∪ ∂wΛn,l. For every group of rectangles R compatible with ξ, with

W (R) ≥ 100m, there exists a disconnecting interval Jc?, d?K such that both (c?, l)

and (d?, l) are in ∂nR, are distance at least m from ∂‖R, and

1

4W (R) ≤ W (R∩ (Jc?, d?K× J0, lK)) ≤ 3

4W (R) .

We pause to comment on why a disconnecting interval with such properties

provides the desired blocks for the block dynamics. The interval Jc?, d?K from the

lemma will be used to define Aint = R ∩ (Jc?, d?K × J0, lK) and Aext = R \ Aint;

their enlargements by m will form the blocks Rint and Rext (see Figure 6.3). The

requirement that W (Aint) be a fraction of W (R) bounded away from 0 and 1 is

so that we only recurse O(log n) times before reaching small enough widths. The

requirement that the corners of Jc?, d?K× J0, lK are a distance at least m from ∂‖R

is so that when we enlarge the sets Aint,Aext by m, we do not overflow beyond the

rectangles containing (c?, l) and (d?, l). Crucially, our ability to pick disconnecting

segments that satisfy this latter requirement will be guaranteed by the compatibility

of R with ξ.

Proof of Lemma 6.1.10. We begin by finding a candidate disconnecting interval

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Jc, dK with (c, l), (d, l) ∈ ∂nR satisfying:

1

3W (R) ≤ W (R∩ (Jc, dK× J0, lK)) ≤ 2

3W (R) . (6.1.2)

In the second part of the proof we show how to modify the interval Jc, dK to obtain

a disconnecting interval Jc?, d?K with the added property that both (c?, l) and (d?, l)

are distance at least m from ∂‖R.

If there exist a pair of vertices (x, l), (y, l) ∈ ∂nR such that 13W (R) ≤ W (R∩

(Jx, yK× J0, lK)) ≤ 23W (R) with (x, l) connected to (y, l) through ξ, then we take

c = x, d = y; that is, we use Jc, dK = Jx, yK as our candidate disconnecting interval.

Suppose otherwise that there does not exist any such boundary connection: then

every pair (x, l), (y, l) ∈ ∂nR connected through ξ is such that

W (R∩ (Jx, yK× J0, lK)) <1

3W (R) , or W (R∩ (Jx, yK× J0, lK)) >

2

3W (R) .

(6.1.3)

If the latter holds, then there is a pair, say (x0, l), (y0, l) ∈ ∂nR, for which the latter

holds with a minimal width. The interval Jx0, y0K would be a disconnecting interval

of wired-type and there is no other vertex (z, l) ∈ ∂nR with z ∈ Jx0 + 1, y0 − 1K

connected to (x0, l) and (y0, l) in ξ since if there were such a z it would violate the

assumption that Jx0, y0K is of minimal width with W (R∩(Jx0, y0K×J0, lK)) > 23W (R)

or that every pair of vertices (x, l), (y, l) ∈ ∂nR connected in ξ satisfy (6.1.3).

Consequently, all other connections through ξ between vertices (x1, l), (y1, l) ∈

∂nR∩ (Jx0 + 1, y0 − 1K× J0, lK) will be such that

W (R∩ (Jx1, y1K× J0, lK)) <1

3W (R) .

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We can then partition the vertices of ∂nR ∩ (Jx0 + 1, y0 − 1K × l) into disjoint

disconnecting intervals of free-wired-type using the following procedure:

1. Let ρ = C1, . . . , Ck be the partition of the vertices of ∂nR∩ (Jx0 + 1, y0 −

1K × l) induced by the boundary condition ξ; every Ci corresponds to a

distinct connected component of ξ;

2. For each Ci, consider the disconnecting interval Li of free-wired-type deter-

mined by the left-most and right-most vertices of Ci in ∂nR∩ (Jx0 + 1, y0 −

1K× l). Notice that some of the Ci’s may be singletons, which we view as

disconnecting intervals of the free-wired-type;

3. Let Li1 , Li2 , . . . , Li` be those disconnecting intervals which are maximal, in

the sense that there does not exist j and k such that Lij ⊂ Lk.

The set of disconnecting intervals Li1 , Li2 , . . . , Li` partitions Jx0 + 1, y0 − 1K into

disjoint disconnecting intervals of free-wired-type with the property that W (R∩

(Lij × J0, lK)) ≤ 13W (R) for every j ∈ 1, . . . , `. We can then use Lemma 6.1.3 to

merge adjacent disconnecting intervals until we obtain a candidate disconnecting

interval Jc, dK ⊂ Jx0, y0K (of free-type), having width W (R ∩ (Jc, dK × J0, lK) ∈

[13W (R), 2

3W (R)].

Now that we have found a candidate disconnecting interval Jc, dK satisfy-

ing (6.1.2), we modify it to obtain a disconnecting interval Jc?, d?K with the property

that both (c?, l) and (d?, l) are distance at least m from ∂‖R.

If (c, l) is at distance at least m from ∂‖R, set c? = c, and similarly if (d, l) is at

distance at leastm from ∂‖R, then set d? = d. Otherwise, suppose (c, l) is at distance

less than m from ∂wRi for some constituent rectangular subset Ri = Jai, biK× J0, lK

of R. Since R is compatible with ξ, the interval Ic = Jbi−1 − m, ai + mK is a

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disconnecting interval, and we set

c? =

ai +m, if Ic is of wired-type, or i = 1, or W (Ri) = 2m;

ai +m+ 1 , if Ic is only of free-type, and W (Ri) > 2m;

note that the first case includes when Ic is of free-wired-type, whereas the second

case only applies when the interval is of free-type and not of wired-type. When

(c, l) is instead at distance less than m from ∂eRi for some i, then we simply set

c? = bi −m.

Symmetrically, if (d, l) is at distance less than m from ∂eRi for some Ri =

Jai, biK× J0, lK, let Id = Jbi −m, ai+1 +mK

d? =

bi −m, if Id is of wired-type, or i = N(R), or W (Ri) = 2m;

bi −m− 1 , if Id is only of free-type, and W (Ri) > 2m.

When (d, l) is at distance less than m from ∂wRi, let d? = ai + m. To see that

this process is well-defined, notice that since W (Ri) ≥ 2m for every i, the points

(c, l), (d, l) cannot be both less than m away from ∂eRi and less than m away from

∂wRi.

We claim that in all of these cases the interval Jc?, d?K is a disconnecting interval.

The fact that (c?, l), (d?, l) ∈ ∂nR are a distance at least m away from ∂‖R follows

directly from the construction.

First, observe that when (c, l), (d, l) are both a distance at least m from ∂‖R,

then we set c? = c and d? = d; in this case Jc?, d?K is disconnecting since Jc, dK was

chosen to be disconnecting.

Next suppose that d was at least a distance m from ∂‖R while c was at distance

less than m from ∂wRi for some i. In this setting, we establish that Jc?, d?K is a

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disconnecting interval by considering the following four cases:

Case 1: i = 1. Note that Ja1 + m, bN(R) −mK and Jc, dK are disconnecting

intervals by compatibility and construction, respectively. Also, W (R∩ (Jc, dK)) ≥100m

3since W (R) ≥ 100m. Hence, c < a1 +m < d < bN(R) −m and Lemma 6.1.5

implies that Jc?, d?K = Ja1 +m, dK is disconnecting.

Case 2: i > 1 and Jbi−1−m, ai +mK is of wired-type. Since W (R∩ (Jc, dK)) ≥100m

3, we have bi−1 − m < c < ai + m < d. Then, by Lemma 6.1.4, Jc, dK is

a disconnecting interval of wired-type, and Lemma 6.1.5 implies that Jc?, d?K =

Jai +m, dK is a disconnecting interval of wired-type.

Case 3: i > 1, Jbi−1 −m, ai +mK is of free-type and W (Ri) > 2m. We again

have bi−1 −m < c < ai + m < d, and by Lemma 6.1.4 Jc, dK is a disconnecting

interval of free-type. Therefore, by Lemma 6.1.5, Jc?, d?K = Jai + m + 1, dK is a

disconnecting interval of free-type.

Case 4: i > 1, Jbi−1 −m, ai + mK is only of free-type and W (Ri) = 2m. In

this case, it must be that i < N(R). Therefore, by the compatibility of R and

ξ, Jbi − m, ai+1 + mK = Jai + m, ai+1 + mK is also a disconnecting interval. In

fact, by Lemma 6.1.4, Jbi −m, ai+1 + mK is a disconnecting interval of free-type.

Applying Lemma 6.1.5 with respect to Jai+m+1, dK and Jai+m, ai+1 +mK implies

Jc?, d?K = Jai +m, dK is a disconnecting interval.

Suppose otherwise that c was at distance less than m from ∂eRi for some i, and d

was still at least a distance at least m from ∂‖R. In this case, W (R∩(Jc, dK)) ≥ 100m3

implies N(R) > 1 as well as i < N(R). Moreover, Jbi − m, ai+1 + mK is a

disconnecting interval by the compatibility of R and ξ. Since bi − m < c <

ai+1 +m < d, Lemma 6.1.4 then implies that when Jbi −m, ai+1 +mK is of wired-

type (resp., of free-type) then Jc, dK is also of wired-type (resp., of free-type), and

169

therefore by Lemma 6.1.5, Jc?, d?K = Jbi − m, dK is a disconnecting interval of

wired-type (resp., of free-type).

The symmetric cases where (c, l) is a distance at least m from ∂‖R, and (d, l)

is a distance less than m from ∂‖R can be checked analogously. The remaining

case in which both (c, l) and (d, l) are a distance less than m from ∂‖R can also be

checked similarly, by first modifying the candidate interval on one side in order to

obtain a disconnecting interval Jc?, dK having (c?, l) a distance at least m from ∂‖R,

and then performing the modification on d to obtain the desired disconnecting

interval Jc?, d?K.

Finally, we claim that in all such situations, Jc?, d?K satisfies

1

4W (R) ≤ W (R∩ (Jc?, d?K× J0, lK)) ≤ 3

4W (R) .

This follows from the facts that W (R) ≥ 100m, |c− c?| ≤ m and |d− d?| ≤ m.

We will now use the disconnecting interval given by Lemma 6.1.10 to define two

subsets Aint and Aext of R, and set Rint and Rext to be their enlargments by m.

Definition 6.1.11. For a group of rectangles R compatible with ξ, let Jc?, d?K be

the disconnecting interval given by Lemma 6.1.10. Then define the interior and

exterior cores as

Aint = R∩ (Jc?, d?K× J0, lK) and Aext = R∩ (Jc?, d?Kc × J0, lK) ;

see Figure 6.3(a). Using the disconnecting interval Jc?, d?K define the interior and

exterior blocks as follows:

1. Let c− = c? −m and c+ = c? +m. Let d− = d? −m and d+ = d? +m.

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Aint AintAext Aext

(a)

Rint Rint

Rext Rext

(b)

Figure 6.3: (a) The cores Aint and Aext and (b) the blocks Rint and Rext. Theblocks Rint and Rext are the enlargements of Aint and Aext by exactly m, and arethus, themselves, groups of rectangles.

2. DefineRint = R∩(Jc−, d+K×J0, lK) andRext = R∩((J0, c+K∪Jd−, nK)×J0, lK);

see Figure 6.3(b).

We will demonstrate that this choice of Rint and Rext has certain fundamental

properties that will facilitate the recursive argument in the proof of Theorem 6.1.1.

Proposition 6.1.12. If R is a group of rectangles compatible with ξ, and moreover,

W (R) ≥ 100m, then the sets Rint and Rext are groups of rectangles satisfying the

following properties:

1. 15W (R) ≤ W (Rint) ≤ 4

5W (R) and likewise 1

5W (R) ≤ W (Rext) ≤ 4

5W (R);

2. Both Rint and Rext are compatible with ξ.

Proof. We show first that Rint and Rext are groups of rectangles (see Defini-

tion 6.1.6). By construction Rint and Rext are unions of rectangular subsets of

Λn,l. Since R is a group of rectangles, every constituent rectangular subset Ri has

W (Ri) ≥ 2m. By Lemma 6.1.10, (c?, l), (d?, l) were such that they are a distance at

least m from ∂‖R; as a consequence Jc−, c+K× J0, lK and Jd−, d+K× J0, lK are subsets

of R. Moreover, every constituent rectangular subset of Rint and Rext is either a

constituent rectangle of R, or contains one of Jc−, c+K× J0, lK and Jd−, d+K× J0, lK,

implying that every rectangular subset of Rint and Rext has width at least 2m.

Together, these verify that Rint and Rext are indeed groups of rectangles.

171

The first property follows from the facts that, by Lemma 6.1.10, 14W (R) ≤

W (Aint) ≤ 34W (R) and 1

4W (R) ≤ W (Aext) ≤ 3

4W (R), while W (R) ≥ 100m,

W (Rint)−W (Aint) = 2m and W (Rext)−W (Aext) = 2m.

To verify the second property, consider Rext first and label its rectangular

subsets Rext1 , ..., Rext

N(Rext) (from left to right) with Rextj = Jaextj , bextj K. Let i, i + 1

be the indices of the two distinct rectangular subsets containing Rext \ Aext. As

before, R1, ..., RN(R) are the constituent rectangular subsets of R. Then, for every

j ∈ 1, ..., N(Rext)− 1 \ i, there is a k ∈ 1, .., N(R)− 1 such that bextj = bk

and aextj+1 = ak+1. Hence, the compatibility of R with ξ guarantees that the interval

Jbextj −m, aextj+1 +mK is disconnecting for every j 6= i.

To see that Jbexti −m, aexti+1 +mK is disconnecting, notice that by construction of

Rext, Jbexti −m, aexti+1 +mK = Jc?, d?K. Finally, since (c?, l), (d?, l) were at distance

at least m from ∂‖R, the interval Jaext1 + m, bextN(Rext) − mK matches the interval

Ja1 +m, bN(R) −mK, and thus by compatibility of R with ξ implies the former is a

disconnecting interval. Altogether these imply the compatibility of Rext with ξ.

Similarly, label the constituent rectangles of Rint as Rint1 , ..., Rint

N(Rint) and notice

that 1 and N(Rint) are the indices of the two rectangles containing Rint \ Aint.

Every interval of the form Jbinti −m, ainti+1 +mK corresponds (up to change of index)

to such an interval for R, so that by compatibility of R with respect to ξ, these

are all disconnecting intervals. The interval Jaint1 +m, bintN(Rint) −mK is exactly the

interval Jc?, d?K given by Lemma 6.1.10, and therefore this is disconnecting by

construction. Together, these imply the compatibility of Rint with ξ.

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6.1.6 Block dynamics coupling time

Here we consider the block dynamics on R with blocks Rint and Rext as defined

in Section 6.1.5; see also Figure 6.3(b). We begin by defining the block dynamics

for a group of rectangles.

Definition 6.1.13. Let R be a group of rectangles with boundary condition ζ.

The block dynamics Xt with blocks B = B1, . . . ,Bk such that Bi ⊂ R and⋃ki=1 E(Bi) = E(R) is the discrete-time Markov chain that at each t, selects i

uniformly at random from 1, ..., k and updates the configuration on E(Bi) from

the stationary distribution conditional on the configuration Xt(Ec(Bi)).

Fix B = Rint,Rext and let gapB(Rζ) be the spectral gap of this block dynamics

on R with boundary condition ζ = (ξ, ωRc), where ξ is a realizable boundary

condition on Λn,l and ωRc is a configuration on Ec(R).

Lemma 6.1.14. Let ξ be a realizable boundary condition on ∂Λn,l that is free on

∂sΛn,l ∪ ∂eΛn,l ∪ ∂wΛn,l and free on vertices in ∂nΛn,l at distance at most m from

∂eΛn,l ∪ ∂wΛn,l. For every q > 1 and p 6= pc(q), there exists K = K(p, q) ≥ 1 such

that for every group of rectangles R compatible with ξ, and every configuration ωRc,

gapB(R(ξ,ω(Rc))) ≥ K−1 .

Proof. We consider the p < pc(q) case first. Let Xt, Yt be two instances of the

block dynamics on R with boundary condition ζ = (ξ, ωRc) started from initial

configurations X0, Y0. We design a coupling P for the steps of Xt and Yt and

bound its coupling time. This yields upper bounds for both the mixing time and

the inverse spectral gap of the block dynamics; see Section 2.2 for a brief overview of

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the coupling method. For this, we will show that for any two initial configurations

X0, Y0

P(X2 = Y2) = Ω(1) . (6.1.4)

Since this bound will be uniform over X0, Y0, we can make independent attempts

at coupling the two chains every two steps. Hence, there would exist T = O(1)

such that

maxX0,Y0

P(XT 6= YT ) ≤ 1

4,

bounding the coupling time by T = O(1) concluding the proof.

First observe that with probability 1/4 the first block to be updated is Rint

and the second is Rext. Suppose this is the case and let us consider the update on

block Rint. Observe that X1(Rint) is distributed according to the random-cluster

measure πθX on Rint, where θX is the boundary condition induced on ∂Rint by

the boundary condition ζ on R and the configuration of X0 in E(R) \ E(Rint).

Likewise, Y1(Rint) has law πθY , with the boundary condition θY defined analogously

but considering the configuration of Y0 in E(R) \ E(Rint) instead. Therefore, any

coupling for the random-cluster measures πθX , πθY yields a coupling for the first

steps of Xt and Yt.

Let θ1 be the boundary condition on ∂Rint induced by ζ and the configuration

that is all wired on E(R) \ E(Rint); let πθ1 be the corresponding random-cluster

measure on Rint. Let Qw, Qe ⊂ Rint be the two rectangles of width m that contain

all the vertices in Rint \ Aint; i.e., Qw ∪ Aint ∪ Qe = Rint, Qw ∩ Aint = ∅ and

Qe ∩ Aint = ∅ (see Figure 6.4(a)). Let ∂E(Qw) be the set edges with one endpoint

in Qw and the other in Aint, and similarly define ∂E(Qe).

174

Aint

Qw Qe

(a)

Aint

(b)

Aint

(c)

Figure 6.4: (a) The block Rint with its subsets Aint, Qw and Qe. (b) The blockRint with the dual-paths (dotted) of a configuration in Γ. (c) The block Rint withthe dual-paths (dotted) of a configuration in Γns

w ∩ Γwew ∩ Γns

e ∩ Γwee ⊂ Γ.

Let Γw be the set of configurations in Rint that have a dual-path in E(Qw) ∪

∂E(Qw) connecting the top-most edge in ∂E(Qw) to an edge in ∂sQw, and similarly

define Γe as the set of configurations inRint that have a dual-path in E(Qe)∪∂E(Qe)

from the top-most edge in ∂E(Qe) to an edge in ∂sQe. (A dual-path is an open

path in the dual configuration.) Let Γ = Γe ∩ Γw; see Figure 6.4(b). The following

lemma supplies the desired coupling.

Lemma 6.1.15. Let q > 1 and p < pc(q). There exists a coupling P1 of the

distributions πθX , πθY , πθ1 such that if (ωθX , ωθY , ωθ1) is sampled from P1, then all

of the following hold:

1. P1(ωθX , ωθY , ωθ1) > 0 only if ωθX ≤ ωθ1 and ωθY ≤ ωθ1;

2. P1(ωθX (Aint) = ωθY (Aint) | ωθ1 ∈ Γ) = 1;

3. There exists a constant ρ = ρ(p, q) > 0 such that P1(ωθ1 ∈ Γ) ≥ ρ.

Hence, if we use the coupling P1 from Lemma 6.1.15 to couple the first step of

the chains, then X1 and Y1 will agree on E(Aint) with probability at least ρ > 0.

175

If this occurs, then we can easily couple the update on Rext in the second step

so that X2 = Y2, since X1(E(Aint)) = Y1(E(Aint)) implies X1(E(R) \ E(Rext)) =

Y1(E(R)\E(Rext)), and thus the boundary conditions induced by the two instances

of the chain on Rext are identical. As a consequence, we obtain that for any X0, Y0,

P(X2 = Y2) ≥ρ4.

which gives (6.1.4) and thus concludes the proof for p < pc(q).

The case when p > pc(q) follows by an analogous dual argument. In this case, the

set Γ has o(1) probability and we therefore replace it by the set Γ∗ = Γ∗e∩Γ∗w, where

Γ∗w is the set of configurations in Rint that have an open path in E(Qw) ∪ ∂E(Qw)

connecting the top-left corner of Aint (c?, l) to ∂sQw; similarly, Γ∗e is the set of

configurations in Rint that have an open path in E(Qe) ∪ ∂E(Qe) connecting the

top-right corner (d?, l) of Aint to ∂sQe. Let θ0 be the boundary condition on ∂Rint

induced by ζ and the all-free configuration on E(R) \ E(Rint); let πθ0 be the

resulting random-cluster distribution on Rint. The constant bound on the coupling

time of the block dynamics would then follow as above from the following dual

analogue of Lemma 6.1.15.

Lemma 6.1.16. Let q > 1 and p > pc(q). There exists a coupling P0 of the

distributions πθX , πθY , πθ0 such that if (ωθX , ωθY , ωθ0) is sampled from P0, then all

of the following hold:

1. P0(ωθX , ωθY , ωθ0) > 0 only if ωθX ≥ ωθ0 and ωθY ≥ ωθ0;

2. P0(ωθX (Aint) = ωθY (Aint) | ωθ0 ∈ Γ∗) = 1;

3. There exists a constant ρ = ρ(p, q) > 0 such that P0(ωθ0 ∈ Γ∗) ≥ ρ.

176

We proceed with the proof of the key Lemma 6.1.15. Parts 1 and 2 of the lemma

follow from a fairly standard coupling technique; see, e.g., [2, 10]. We shall also use

this approach later in the proof of Lemma 6.2.5. Part 3 will be a consequence of

the EDC property (2.1.1); see proof of Claim 6.1.17.

Proof of Lemma 6.1.15. Let

L = ∂wQw ∪ ∂nQw ∪ ∂eQe ∪ ∂nQe;

note that L ∩ Aint = ∅. For an FK configuration ω on Rint let

F (ω) := Rint \⋃

v∈LC(v, ω) ,

where C(v, ω) is the vertex set of the connected component of v in ω, ignoring the

boundary connections.

Clearly, πθ1 πθX and πθ1 πθY and thus there exist monotone couplings PX

(resp., PY ) for πθX and πθ1 (resp., πθY and πθ1). We use PX and PY to construct

the coupling P1 as follows:

1. sample (ωθX , ωθ1) from PX and sample ωθY from PY ( · | ωθ1 );

2. if Aint ⊆ F (ωθ1), sample ω∆ from πη1

∆ —where ∆ = ∆(ωθ1) is the subgraph

induced by F (ωθ1) and η1 is the boundary condition on ∂F (ωθ1) induced by θ1

and the configuration of ωθ1 on E(Rint)\E(F (ωθ1))—and make ωθ1(F (ωθ1)) =

ωθX (F (ωθ1)) = ωθY (F (ωθ1)) = ω∆.

Let P1 be the resulting distribution of (ωθX , ωθY , ωθ1). After step (i), P1 has the

desired marginals. Moreover, we claim that replacing the configuration in F (ωθ1)

177

with ω∆ in step (ii) has no effect on the distribution, provided Aint ⊆ F (ωθ1). For

this, we show that the three boundary conditions η1, ηX , ηY induced on ∂F (ωθ1)

by the configurations of ωθX , ωθY , ωθ1 on E(Rint) \ E(F (ωθ1)), respectively, and

the corresponding boundary conditions θX , θY , θ1 are all the same.

First, observe that the boundary condition on ∂sAint is, in all three cases, free

by assumption. Also, from the definition of F (ωθ1) it follows that every edge of

E(Rint) \ E(F (ωθ1)) incident to ∂F (ωθ1) is closed in ωθ1 ; hence the same holds

for ωθX and ωθY . The remaining portion of ∂F (ωθ1) is precisely the set of vertices

(∂Aint ∩ ∂R) \ ∂sR. To show that η1, ηX , ηY also agree on (∂Aint ∩ ∂R) \ ∂sR we

use the fact that top-left and top-right corners of Aint correspond to the endpoints

of the disconnecting interval Jc?, d?K. Indeed, for the boundary conditions η1, ηX , ηY

to disagree on (∂Aint ∩ ∂R) \ ∂sR it must be the case that there are at least two

distinct connected components of ζ = (ξ, ωRc) connecting (∂Aint ∩ ∂R) \ ∂sR and

∂R \ ∂Aint. Since (c?, l), (d?, l) /∈ ∂‖R and ξ is free on ∂sR, this would require at

least two distinct connected components of ξ connecting vertices in Jc?, d?K× l

to vertices in Jc?, d?Kc × l. However, when Jc?, d?K is a disconnecting interval of

free-type, there are no such connected components, and when it is disconnecting of

wired-type, the planarity of realizable boundary conditions implies that there can

be at most one such connected component, which is exactly the component of ξ

containing both (c?, l) and (d?, l).

Altogether, these together imply that when Aint ⊆ F (ωθ1), the three boundary

conditions η1, ηX , ηY induced on ∂F (ωθ1) are the same. The domain Markov

property of random-cluster measures (see, e.g., [47]) then guarantees that replacing

the configuration in F (ωθ1) with ω∆ had no effect on the distributions.

Finally, note that if ωθ1 ∈ Γ, then the vertices in the boundary components of

178

L, i.e.,⋃v∈LC(v, ωθ1), will be confined to Qw ∪Qe, in which case Aint ⊆ F (ωθ1).

This establishes parts 1 and 2, and part 3 follows directly from the following claim,

which will conclude the proof.

Claim 6.1.17. Let q > 1 and p < pc(q). There exists ρ = ρ(p, q) > 0 such that

πθ1(Γ) ≥ ρ.

Proof. Let Γnsw (resp., Γns

e ) be the set of configurations on Rint such that there is

dual-path from ∂nQw to ∂sQw (resp., from ∂nQe to ∂sQe) in E(Qw)∪∂E(Qw) (resp.,

E(Qe)∪∂E(Qe)). Also, let Γwew (resp., Γwe

e ) be the set of configurations onRint such

that there is dual-path in E(Qw)∪∂E(Qw) (resp., in E(Qe)∪∂E(Qe)) between the

left-most edge in ∂nQw (resp., the right-most edge in ∂nQe) and the top-most edge

of ∂E(Qw) (resp., ∂E(Qe)); see Figure 6.4(c). Note that Γnsw ∩Γwe

w ∩Γnse ∩Γwe

e ⊂ Γ.

The width of Qw is m = C? log l. Hence, the EDC property (2.1.1) implies

that when C? is large enough there exists a constant ρ0 = ρ0(p, q) > 0 such that

πθ1(Γnsw ) ≥ ρ0 and similarly for Γns

e . The EDC property (2.1.1) also implies that

πθ1(Γwew ) ≥ ρ1 and πθ1(Γwe

e ) ≥ ρ1, for a suitable ρ1 = ρ1(p, q) > 0. We justify this

as follows. By the EDC property (2.1.1), there exists some constant D such that

with probability Ω(1), no pair of vertices whose distance is at least D, one of which

is in ∂nQw \ ∂wQw and the other in ∂eQw ∪ ∂sQw ∪ ∂eQw, will be connected in Qw.

At the same time, with probability Ω(1), we can force O(D) edges bordering those

pairs of vertices that are distance less than D to be closed so that no pair of vertices

that are closer than D are connected in Qw either. The analogous reasoning holds

for Γwee and Qe. Since each of the events Γns

w ,Γwew ,Γns

e ,Γwee are decreasing events,

by the FKG inequality (see, e.g., [47]) we get

πθ1(Γ) ≥ πθ1(Γnsw ∩ Γwe

w ∩ Γnse ∩ Γwe

e ) ≥ πθ1(Γnsw )πθ1(Γwe

w )πθ1(Γnse )πθ1(Γwe

e ) ≥ ρ20ρ

21,

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and thus we can take ρ = ρ20ρ

21 to have the desired estimate.

Proof of Lemma 6.1.16. The proof of Lemma 6.1.15 carries to Lemma 6.1.16

with certain natural modifications we describe next. As before, we let L = ∂wQw ∪

∂nQw∪∂eQe∪∂nQe and let (L∗, E(L∗)) be the dual-graph induced by the set of dual-

edges intersecting E(L) in (Z2)∗; its vertex set consists of exactly 2E(L)−1 vertices,

and we refer to those outside of Rint as ∂extL∗. Similarly, let (R∗int, E(R∗int)) and

(A∗int, E(A∗int)) be the dual-graphs induced by the dual-edges intersecting E(Rint)

and E(Aint) in (Z2)∗ respectively.

For an FK configuration ω on E(Rint) we define a dual version of F (ω) as

F ∗(ω) := R∗int \⋃

v∗∈∂extL∗C∗(v∗, ω) ,

where C∗(v∗, ω) is the dual-vertex set of the connected component of v∗ in the

dual-configuration ω∗ (ignoring the boundary connections).

Using monotone couplings for πθ0 and πθX , and for πθ0 and πθY , we can define P0

analogously to P1 with the configuration on E(F ∗(ωθ0)) being resampled whenever

A∗int ⊆ F ∗(ωθ0). Observe that updating the dual configuration on E(F ∗(ωθ0)) is

equivalent to updating the primal edges intersecting E(F ∗(ωθ0)). The fact that

resampling the configuration in E(F ∗(ωθ0)) has no effect on the distribution when

A∗int ⊆ F ∗(ωθ0) follows in similar fashion to the proof of Lemma 6.1.15. Indeed,

from the definition of F ∗(ωθ0), when A∗int ⊆ F ∗(ωθ0) there is a primal connection

between (c?, l) and ∂sQw in E(Qw) ∪ ∂E(Qw) together with a primal connection

between (d?, l) and ∂sQe in E(Qe) ∪ ∂E(Qe), so that A∗int ⊆ F ∗(ωθ0) implies the

event Γ∗. Together with the fact that Jc?, d?K is a disconnecting interval and the

assumptions that ξ is free on ∂sAint and (c?, l), (d?, l) /∈ ∂‖R, this ensures that the

180

three induced boundary conditions on E(F ∗(ωθ0)) coincide.

Finally, part 3 of the lemma follows by an analogous argument to that in

Claim 6.1.17, only replacing the EDC property by the matching exponential decay

of dual-connectivities when p > pc(q).

6.1.7 Proof of Theorem 6.1.1

In this section, we put together the results from Sections 6.1.3–6.1.6 to prove

Theorem 6.1.1. We first remind the reader of the following spectral gap estimate for

the FK Glauber dynamics. This proposition is written in the spin system setting

in [67] as presented in Theorem 2.2.8, but the proof follows mutatis mutandis for

the FK model, and the proof is therefore omitted. Also, we note that this theorem

holds in more generality for arbitrary graphs with arbitrary boundary conditions,

but for clarity we choose to state it here for groups of rectangles.

Theorem 6.1.18 ([67, Proposition 3.4]). Consider the FK Glauber dynamics on

a group of rectangles R with boundary condition ζ. Let gaprc(Rζ) and gapB(Rζ),

respectively, be the spectral gaps of the FK Glauber dynamics on R and of the block

dynamics with blocks B = R1, . . . ,Rk such that Ri ⊂ R and⋃ki=1E(Ri) = E(R).

For every p, q there exists γ = γ(p, q) ∈ (0, 1) such that

gaprc(Rζ) ≥ γ · ( maxe∈E(R)

#i : E(Ri) 3 e)−1 · gapB(Rζ) · mini=1,...,kη∈Ω(Rci )

gaprc(R(ζ,η)i ) ,

where Ω(Rci) denotes the set of FK configurations on E(R) \ E(Ri).

The final ingredient is the following mixing time estimate for the base case in

our recursive proof.

181

Lemma 6.1.19. Consider a group of rectangles R0 ⊂ Λn,l with W (R0) ≤ 100m.

For every q > 1 and p 6= pc(q), there exists κ = κ(p, q) > 0 such that for every

boundary condition ζ on R0,

gaprc(Rζ0) ≥ 1

l(log l)2 · qκl.

Proof. Note that |∂R0| = O(m + l). Hence, we can first modify the boundary

conditions to be all free on all of ∂R0, incurring a cost of a qO(l) factor in the

spectral gap by Eq. (2.2.9) and Lemma 2.2.13; recall that m = O(log l). Then we

can use the fast mixing result of [10], for instance, to bound the mixing time on R0

with free boundary condition by O(l(log l)2). This translates into a lower bound

for the spectral gap and the result follows.

Proof of Theorem 6.1.1. Fix q > 1, p 6= pc(q) and Λn,l with a realizable bound-

ary condition ξ′ that is free on ∂eΛn,l ∪∂sΛn,l ∪∂wΛn,l. By Eqs. (2.2.9)–(2.2.10), we

may modify ξ′ to a boundary condition ξ that is also free on all vertices a distance

at most m = C? log l from ∂eΛn,l ∪ ∂wΛn,l at a cost of an exponential in m factor

in the mixing time of the FK-dynamics. Let ξ be the resulting realizable boundary

condition.

We wish to prove, by induction, that for every 100m ≤ s ≤ n, every group of

rectangles Rs ⊂ Λn,l that is compatible with ξ and has W (Rs) = s satisfies

gap(R(ξ,ωRcs )s

)≥ 1

l(log l)2qκl · blog s(6.1.5)

for some b = b(p, q) > 0 to be chosen, uniformly over all configurations ωRcs on

Ec(Rs). Eq. (6.1.5) concludes the proof since Λn,l is a group of rectangles with

182

W (Λn,l) = n and is compatible with ξ.

The base case of this induction was shown in Lemma 6.1.19. Now suppose

inductively that this holds for all 1 ≤ k ≤ s − 1 for some s ≤ n; we show that

it also holds for s. Fix any Rs that is compatible with ξ, and any configuration

ωRcs . Then, if we let Rint = Rint(Rs) and Rext = Rext(Rs) be the blocks given

by Definition 6.1.11 and Bs the block-dynamics with respect to these blocks, by

Theorem 6.1.18

gaprc

(R(ξ,ωRcs )s

)≥ γ

2· gapBs

(R(ξ,ωRcs )s

)· mini∈int,ext

minωRc

i

gaprc

(R

(ξ,ωRci)

i

)≥ γ

2K· mini∈int,ext

minωRc

i

gaprc

(R

(ξ,ωRci)

i

),

where the second inequality follows from Lemma 6.1.14. By Proposition 6.1.12,

maxW (Rint),W (Rext) ≤ 45s, and we can apply the inductive hypothesis to bound

the second term on the right-hand side above. Combined with Lemma 6.1.19, we

see that the choice of b = (2γ−1K)1

log(5/4) ensures that (6.1.5) holds also for Rs.

(Note that 2γ−1K ≥ 1.)

This establishes the result for the case when the boundary condition is free on

∂eΛn,l ∪ ∂sΛn,l ∪ ∂wΛn,l. As noted earlier (see Remark 2.2.1), this implies by duality

the same bound for the class of realizable boundary conditions ξ that are wired on

∂eΛn,l ∪ ∂sΛn,l ∪ ∂wΛn,l for all p 6= pc(q).

183

Csw Cse

Cnw Cne

5r

5r

(a)

Rs

Rn

Rw Ren−6r

3r

(b)

e

e

2r+1

(c)

Figure 6.5: (a) The subsets Cne, Cnw, Cse, and Csw. (b) The subsets Rn, Re, Rw

and Rs. (c) B(e, r) for two edges e of Λn.

6.2 Polynomial mixing time for realizable bound-

ary conditions

In this section we prove Theorem 5. This theorem is proved for p < pc(q) using

the technology introduced in Section 6.0.1; namely, we construct a collection of

subsets B for which we can establish LM and MSM; see Definitions 6.0.1–6.0.3.

To establish LM we crucially use Theorem 6.1.1. The results for p > pc(q) follow

from the planar duality of the model and the self-duality of the class of realizable

boundary conditions, as explained in Section 2.2.1, Remark 2.2.1.

For general realizable boundary conditions, proving LM for a collection of

subsets B for which MSM holds is the main challenge. This is because, for MSM

to hold for a collection B for all realizable boundary conditions, a subset in B

needs to contain Ω(n) edges. In particular, some element of B must include most

(or all) edges near ∂Λn, as otherwise it is straightforward to construct examples

of realizable boundary conditions for which MSM does not hold. Thus, a trivial

(exponential in the perimeter) upper bound for the mixing time on those subsets

with Ω(n) edges would be unhelpful and we use Thoerem 6.1.1.

184

We now define the collection of blocks for which we can establish both LM

and MSM. Let r ∈ N and let Cne, Cnw, Cse, Csw ⊂ Λn be the four square boxes

of side length 5r with a corner that coincides with a corner of Λn; see Figure

6.5(a). Let Rn ⊂ Λn be the (n− 6r)× 2r rectangle at distance 3r from both ∂wΛn

and ∂eΛn whose top boundary is contained in ∂nΛn and let Re, Rw, Rs be defined

analogously; see Figure 6.5(b). Let R = Rn ∪Re ∪Rw ∪Rs. Now, for e ∈ E(Λn),

let B(e, r) ⊂ Λn be the set of vertices in the minimal square box around e such

that d(e,Λn \ B(e, r)) ≥ r. Note that if d(e, ∂Λn) > r, then B(e, r) is just a

square box of side length 2r + 1 centered at e; otherwise B(e, r) intersects ∂Λn; see

Figure 6.5(c). Finally, let

Br = Cne, Cnw, Cse, Csw, R ∪ B(e, r) : e ∈ E(Λn), d(e, ∂Λn) > r. (6.2.1)

We claim that LM holds for Br with r = Θ(log n) and T = O(nC) for some

constant C > 0.

Theorem 6.2.1. Let q ≥ 1, p < pc(q) and r = c0 log n with c0 > 0 independent of

n. There exists a constant C > 0 such that LM holds for every realizable boundary

condition ξ and Br with T = O(nC).

The subsets B(e, r) in Br and the corner boxes Cne, Cnw, Cse and Csw are small

enough that crude bounds for their mixing times are sufficient. As mentioned

earlier, the main challenge for proving local mixing for Br is to derive a mixing time

bound for R = Rn ∪Re ∪Rw ∪Rs as it intersects the boundary of Λn and contains

Ω(n) vertices. To establish such a bound we rely on Theorem 6.1.1. In particular,

we relate the mixing time of the FK-dynamics on R to that of the FK-dynamics

on a single thin rectangle by concatenating the four rectangles constituting R, one

185

after another, such that the union of their outer boundaries make up the northern

boundary of the new rectangle.

The final ingredient of the proof is establishing MSM for the collection Br.

We show that MSM holds for Br with r = Θ(log n) for all realizable boundary

conditions ξ where the vertices in ∂Λn at distance 5r from the corners of Λn are

free in ξ. This is sufficient since any realizable boundary condition can be turned

into a realizable boundary condition with this property by simply removing all

connections in ξ involving vertices near the corners of Λn; this modification can

change the mixing time of the FK-dynamics by a factor of at most exp(O(r)); see

Eqs. (2.2.9)–(2.2.10). Theorem 5 then follows from Theorems 6.2.1, 6.2.2 and 6.0.5.

Theorem 6.2.2. Let q ≥ 1, p < pc(q) and r = c0 log n with c0 > 0 independent

of n. Let ξ be a realizable boundary condition with the property that every vertex

v ∈ ∂Λn at distance at most 5r from a corner of Λn is free in ξ. Then, for all

sufficiently large c0 > 0, MSM holds for ξ and Br with δ < 1/(12|E(Λn)|).

We are now ready to prove Theorem 5 using the above.

Proof of Theorem 5. As mentioned earlier and in Remark 2.2.1, by duality of

the dynamics and self-duality of the class of realizable boundary conditions, it

suffices to prove the theorem for p < pc(q). Let P be the set of realizable boundary

conditions of Λn = (Λn, E(Λn)). For η ∈ P , let (η1, η2, . . . , ηk) denote the partition

of ∂Λn corresponding to η, and let η(`) be the boundary condition obtained as

follows: for each v ∈ ∂Λn, if v ∈ ηi and v is at distance at most ` from a corner of

Λn, remove v from ηi and add it as a singleton to the partition. Let P` be the set

of all boundary conditions obtained in this manner.

Consider Λn with arbitrary realizable boundary conditions ξ ∈ P. By (2.2.9)–

186

(2.2.10), we see that there exists C > 0 such that for every ξ ∈ P , we have

tmix(Λξn) ≤ Cq8C` · n2 · tmix(Λξ(`)

n ) .

It therefore suffices to prove the mixing time estimate uniformly over all modified

boundary conditions η ∈ P` for ` = 5r and r = c0(log n) with c0 taken to be

sufficiently large, as q8C` would only be polynomial in n. By Theorem 6.2.2, for c0

large enough, uniformly over all such boundary conditions we have moderate spatial

mixing with respect to Br and η ∈ P` with δ < 1/(12|E(Λn)|). Theorem 6.2.1

implies that LM holds for Br and η ∈ P` with T = O(nc) where c > 0 constant.

The result then follows from Theorem 6.0.5.

Remark 6.2.3. We note that Theorem 5 also holds for the FK-dynamics on

rectangles Λn,l ⊂ Z2 with ` ≤ n, provided these rectangles are not too thin. For

example, when l = Ω((log n)2), our proofs would yield that the mixing time of the

FK-dynamics is polynomial in n.

6.2.1 Local mixing for realizable boundary conditions

In this subsection, we prove Theorem 6.2.1. As mentioned earlier, this theorem

may be viewed as a corollary of Theorem 6.1.1, which bounds the mixing time of

the FK Glauber dynamics on thin rectangles.

Proof of Theorem 6.2.1. Let r = c0 log n. We wish to show that each of the

subsets in Br has mixing time O(nc) under the boundary conditions (1, ξ) and (0, ξ).

We begin by bounding the mixing time on the square boxes Cne, Cnw, Cse, Csw

and B(e, r) of Br. Since these have side length O(log n), a crude estimate on the

mixing time is sufficient. For instance, by (2.2.9), at a cost of exp(O(r)) = nO(1)

187

factor, we can compare the mixing time in these boxes with boundary condition

either (1, ξ) or (0, ξ) to the mixing time on equally sized boxes with free boundary

conditions. In this setting, an upper bound of O((log n)2 log log n) is known [9],

and thus we obtain an nO(1) bound for their mixing times.

It remains to bound the mixing time of the FK dynamics on the set R =

Rn ∪ Re ∪ Rw ∪ Rs. For this, we use Theorem 6.1.1. We argue that the mixing

time of the FK dynamics on R is roughly equal to that of the FK dynamics on a

[4(n− 6r)− 3]× 2r rectangle Q with a suitably chosen boundary condition. We

proceed to construct the rectangle Q and a boundary condition ξ′ whose vertices,

edges and wirings are identified with those of R and (1, ξ). The case of R and (0, ξ)

is handled later in similar fashion.

We introduce some notation first. For a rectangle S, let Sα denote the rectangle

that results from a clockwise rotation of S by an angle of amplitude α. Also, if

S1, . . . , Sk are rectangles of the same height, let [S1, . . . , Sk] denote the rectangle

obtained by identifying the vertices in ∂eSi with those in ∂wSi+1 for all i =

1, . . . , k− 1. When identifying the vertices, the double edges are removed. We take

Q =[Rπ/2w , Rn, R

−π/2e , R−πs

].

Observe that every vertex of Q, except those where the boundary overlaps occur,

correspond to exactly one vertex in R; vertices in the overlaps correspond to exactly

two vertices in R. Conversely, every vertex in R corresponds to exactly one vertex

of Q. The edges of Q and R are identified using this correspondence between the

vertices. Observe also that ∂nQ = ∂R∩ ∂Λn. We construct the boundary condition

ξ′ of Q as follows. If u, v ∈ ∂R ∩ ∂Λn are wired in ξ, the corresponding vertices are

188

also wired in ξ′. The boundary condition ξ′ is also wired along ∂wQ ∪ ∂nQ ∪ ∂eQ.

Claim 6.2.4. The boundary condition ξ′ of Q is realizable. In particular, ξ′ can

be realized by an FK configuration in the half plane of Z2 containing only vertices

north of ∂nQ, and a wiring of ∂eQ ∪ ∂sQ ∪ ∂wQ.

Finally, to completely capture the effect of the boundary condition (1, ξ) on R,

each of the three columns in Q that corresponds to overlaps of columns from R,

are externally wired.

Now, by Theorem 6.1.1 and (2.2.9), we have

gaprc(Qξ′) = n−O(1) .

We claim next that the FK Glauber dynamics on R with boundary condition

(1, ξ) has roughly the same gap as the FK Glauber dynamics on Q with boundary

condition ξ′. To see this, we add a double edge to each edge of Q that corresponds

to two edges in R. With this modification, there is now a one-to-one correspondence

between the FK configurations in R and Q. Also, adding these edges has almost no

effect on the mixing time of the FK dynamics in Q, as their endpoints are wired,

and so they only need to be updated once to mix. Moreover, by construction, the

boundary condition ξ′ for Q together with the wiring of the overlapping columns

in Q encode exactly the same connectivities as the boundary condition (1, ξ) for

R. Hence, for every pair of FK configurations on Q, the FK dynamics has the

same transition probability as FK dynamics on R between the corresponding

configurations. Consequently, we can conclude that

gaprc(R(1,ξ)) = n−O(1) .

189

Finally, for the case of the FK dynamics on R with boundary condition (0, ξ)

we can simply wire ∂wRn to ∂nRw, ∂sRw to ∂wRs, ∂eRs to ∂sRe and ∂nRe to ∂eRn,

which only incur a penalty of nO(1) by (2.2.9) and proceed as in the previous

case.

Proof of Claim 6.2.4. First note that ∂nQ corresponds to ∂R ∩ ∂Λn. Let ω be

an FK configuration on Z2 \ Λn that realizes ξ. A path from u ∈ ∂R ∩ ∂Λn to

v ∈ ∂R∩∂Λn in ω splits ∂R∩∂Λn into two parts R1 and R2, one containing all the

vertices from u to v in ∂R∩∂Λn clockwise and the other all the vertices from u to v

in ∂R∩∂Λn counterclockwise. The planarity of Z2 implies that any other boundary

component will be either completely contained in R1 or R2. From this property, it

follows that if v1, v2 ∈ ∂nQ are wired in ξ′, then Jv1, v2K is a disconnecting interval.

This implies that the connectivities of ξ′ in ∂nQ can be realized by a configuration

on the half plane of Z2 that contains all the vertices north of ∂nQ. For example,

every component C = c0, . . . , ck of ξ, with ci to the left of ci+1, can be realized

by the gadget consisting k paths of length hC starting at c0, . . . , ck and going north,

together with one path parallel to ∂nQ that joins the endpoints of all of these path.

Since Jci, ci+1K is a disconnecting interval for all i and C, we can choose hC for each

C so that the resulting configuration is a valid configuration in the half plane.

6.2.2 Moderate spatial mixing for realizable boundary con-

ditions

In this section we prove Theorem 6.2.2. We reduce the moderate spatial mixing

condition (6.0.1) to bounding the probability of certain connectivities in an FK

configuration. Specifically, if e ∈ S ⊂ Λn, the configuration on Ec(S) affects the

190

state of e when there are paths from e to the boundary of S; the probability of such

paths is maximized when we assume an all wired configuration on Ec(S). Recall

that for S ⊂ Λn, we let Sc = Λn \ S, and we use Ec(S) = E(Λn) \ E(S).

Lemma 6.2.5. Consider the FK model on Λn with arbitrary boundary condition

ξ on ∂Λn. For any e ∈ E(Λn), any S ⊂ Λn such that e ∈ E(S), and any pair of

configurations ω1, ω2 on Ec(S):

∣∣∣πξΛn( e = 1 | Ec(S) = ω1 )− πξΛn( e = 1 | Ec(S) = ω2 )∣∣∣

≤ πξΛn

(e ξ←→ ∂S \ ∂Λn | Ec(S) = 1

),

where e ξ←→ ∂S denotes the event that there is a path from e to ∂S taking into

account the connections induced by ξ.

In the proof of Theorem 6.2.2 we use this lemma; its proof via machinery from [2]

will be straightforward.

Proof of Theorem 6.2.2. We need to show that as long as c0 is large enough,

for every e ∈ E(Λn) there exists Be ∈ Br such that (6.0.1) holds for some δ <

1/(12|E(Λn)|). For each e ∈ E(Λn) the subset Be is chosen as follows:

1. If d(e, ∂Λn) > r, then Be = B(e, r);

2. Otherwise, if e ∈ R and d(e, ∂R \ ∂Λ) ≥ r, then Be = R;

3. Otherwise, e ∈ Ci for some i ∈ ne,nw, sw, se, and we take Be = Ci.

191

By Lemma 6.2.5, for every e ∈ E(Λn),

∣∣∣πξΛn( e = 1 | Ec(Be) = 1 )− πξΛn( e = 1 | Ec(Be) = 1 )∣∣∣

≤ πξΛn

(e

ξ←→ ∂Be \ ∂Λn | Ec(Be) = 1).

In all three cases above, by construction d(e, ∂Be \ ∂Λn) ≥ r. This together

with the fact that all vertices of ∂Λn within distance 5r from the corners have no

connections in ξ, implies that for e to be connected to ∂Be \ ∂Λn a path of open

edges reaching a distance at least r is required in the configuration on Be. The

EDC property (see (2.1.1)) implies that for c0 large enough

πξΛn

(e η←→ ∂Be | Ec(Be) = 1

)≤ 1

12|E(Λn)|,

and the result follows.

We conclude this section with the proof of Lemma 6.2.5.

Proof of Lemma 6.2.5. Let (1, ξ) be the boundary condition induced on S by

ξ and the event Ec(S) = 1. Similarly, let θ1 (resp., θ2) be the boundary

condition induced on S by configurations ω1 (resp., ω2) on Ec(S) and ξ. For

ease of notation set πθ1 = πξΛn(· | Ec(S) = ω1), πθ2 = πξΛn(· | Ec(S) = ω2) and

π(1,ξ) = πξΛn(· | Ec(S) = 1). For an FK configuration ω on S let

Γ(1,ξ)(ω) := S \⋃

v∈∂S\∂ΛnC(v, ω) ,

where C(v, ω) is the set of vertices in the connected component of v in ω, taking into

account the connectivities induced by (1, ξ). In words, Γ(1,ξ)(ω) is the set of vertices

192

of S not connected to ∂S \ ∂Λn in ω using possibly the boundary connections.

We claim that there exists a coupling P of the distributions πθ1 , πθ2 and π(1,ξ)

such that P(ω1, ω2, ω) > 0 only if ω1 ≤ ω and ω2 ≤ ω on E(S) and ω1, ω2 agree on

all edges with both endpoints in Γ(1,ξ)(ω). Given this coupling P, we have

∣∣πξΛn( e = 1 | Ec(S) = ω1 )− πξΛn( e = 1 | Ec(S) = ω2 )∣∣

≤ P(e /∈ E(Γ(1,ξ)(ω))

)= πξΛn

(e

ξ←→ ∂S \ ∂Λn | Ec(S) = 1),

as claimed. The construction of the coupling P is standard and is thus ommitted;

see, e.g., [2, 9] and the proof Lemma 6.1.15 for similar constructions.

6.3 Near-optimal mixing time for typical bound-

ary conditions

In this section we provide the proof of Theorem 6, where we establish a sharper

O(n2) mixing time upper bound for the FK Glauber dynamics on Λn = (Λn, E(Λn))

for the class of boundary conditions we call typical.

Definition 6.3.1. Let ω be a random-cluster configuration on Z2, and let ξω be the

boundary condition on ∂Λn induced by the edges of ω in E(Z2) \ E(Λn). Suppose

ω is sampled from πZ2,p,q. A set C of realizable boundary conditions for Λn is called

typical (with respect to (p, q)) if ξω ∈ C with probability 1− o(1).

We define classes of boundary conditions Cα and C?α, consisting of realizable

boundary conditions whose distinct boundary components consist only of vertices

193

at most distance α log n apart in ∂Λn. For any realizable boundary condition ξ

corresponding to a partition ξ1, ξ2, ... of ∂Λn, let L(ξi) be the smallest connected

subgraph of ∂Λn containing all vertices in ξi.

Definition 6.3.2. We say a boundary condition ξ on ∂Λn with corresponding

partition ξ1, ξ2, ... is in Cα if maxi |L(ξi)| ≤ α log n. We say that a realizable

boundary condition ξ is in C?α if its dual boundary condition ξ? is in Cα; see Section

2.1.2 for the definition of dual configuration. We refer to the classes Cα and C?α as

α-localized boundaries.

It is straightforward to see that if one samples a “random” boundary condition

from the infinite-volume measure πZ2,p,q, then with high probability, the induced

boundary condition on ∂Λn is α-localized for some α > 0. Since there is a unique

random-cluster measure on Z2 when p 6= pc(q), this is well-defined.

Lemma 6.3.3. For every q ≥ 1 and p < pc(q), the class of boundary conditions

Cα is typical with respect to (p, q) for sufficiently large α > 0. Similarly, for every

q ≥ 1 and p > pc(q), the class C?α is typical with respect to (p, q) for sufficiently

large α > 0.

Proof. By planar duality (namely the duality of the sets of boundary conditions

Cα and C?α, it suffices to prove the case p < pc(q)). For any u, v ∈ Z2, by the EDC

property (2.1.1), we have that for q ≥ 1 and p < pc(q) there exists c = c(p, q) > 0

such that πZ2(u ↔ v) ≤ e−cd(u,v) . Let u, v ∈ ∂Λn and suppose d(u, v) ≥ α log n.

Then, there exists some C(p, q) > 0 such that for sufficiently large α > 0,

πZ2\Λn(uZ2\Λn←→ v) = πZ2(u

Z2\Λn←→ v) ≤ πZ2(u↔ v) ≤ Ce−cα logn ≤ 1

n3,

194

where recall that uZ2\Λn←→ v denotes the event that there exists a path from u to

v in Z2 \ Λn. A union bound over all pairs of vertices in ∂Λn implies that if ω is

sampled from πZ2 and ξω is the resulting boundary condition on ∂Λn, then ξω ∈ Cα

with probability 1− o(1) and thus Cα is typical.

Remark 6.3.4. One may also be interested in the following notion of typicality,

which sometimes comes up in recursive mixing time upper bounds. Let q ≥ 1 and

p < pc(q) (resp., p > pc(q)) and consider a random-cluster sample from πζR2n,p,q,

where R2n is the concentric box of side length 2n containing Λn with arbitrary

boundary condition ζ. One could easily show that the boundary condition induced

by the configuration on R2n \Λn is in Cα (resp., C?α) with probability 1− o(1). This

would follow by coupling this measure to the infinite-volume measure using the fact

that Cα is a decreasing event, and finding a dual circuit in the annulus R2n \ Λn

(which exists with probability 1−O(e−Ω(n))).

Theorem 6 follows immediately from the following theorem proving near-optimal

mixing for α-localized boundary conditions, together with Lemma 6.3.3.

Theorem 6.3.5. For every q > 1, p < pc(q) (resp., p > pc(q)), and every α > 0,

there exists a constant C > 0 such that for every realizable boundary condition

ξ ∈ Cα (resp., ξ ∈ C?α) on ∂Λn, the mixing time of the discrete-time FK Glauber

dynamics on the n× n box Λn with boundary condition ξ is O(n2(log n)C).

In particular, we prove Theorem 6.3.5 in the regime p < pc(q) and ξ ∈ Cα and

this translates to a matching bound at p > pc(q) and ξ ∈ C?α by duality. To prove

this theorem we again use the general framework from Theorem 6.0.5. Namely,

we construct a collection of subsets of Λn for which we can establish MSM and

LM; see Definitions 6.0.1 and 6.0.3. The fact that ξ ∈ Cα will allow us to prove

195

e

∂Λ

B(e, r′)

B(e, r)

· · · · · ·

Figure 6.6: If r = Θ((log n)2) and r′ = Θ(log n), influence from outside of B(e, r′)may be easily propagated to e through long boundary connections in B(e, r′); butto propagate influence from the exterior of B(e, r), Ω(log n) of them would have tobe connected in Λn.

MSM with respect to Θ((log n)2)×Θ((log n)2) rectangles along the boundary, and

Theorem 5 will provide the LM estimate on these rectangles.

Consider the collection Br = B(e, r) : e ∈ E(Λn). Recall that for r ≥ 0 and

e ∈ E(Λn), we set B(e, r) ⊂ Λn to be the set of vertices in the minimal square box

around e such that d(e,Λn \B(e, r)) ≥ r; see Figure 6.5(c). We first show that

MSM holds for Br and ξ ∈ Cα when r = Θ((log n)2) and δ < n−3.

Lemma 6.3.6. Let q ≥ 1, p < pc(q), α > 0, η ∈ Cα, r = c0(log n)2 and B =

B(e, r) : e ∈ E(Λn). For large enough c0 > 0, MSM holds for η, B for some

δ < n−3.

For this lemma, it is crucial that r = Θ((log n)2), as MSM does not hold for typical

boundary conditions for Br when, for example, r = Θ(log n). This is because in a

typical configuration ω on Z2 \Λn it is likely that there exist pairs of vertices of ∂Λn

at distance γ log n, for a suitably small constant γ > 0, that are connected in ω.

Thus, for some e ∈ E(Λn) close to ∂Λn, it is possible for the configuration outside of

B(e, r) to exert a strong influence on the state of e when r = γ′ log n with constant

γ′ > 0, even if γ′ γ; the presence of a constant number of open edges on (or near)

∂Λn would propagate the influence from Λn \B(e, r) to e. Taking r = Ω((log n)2)

196

avoids this issue, since, roughly speaking, Ω(log n) open edges at specific points in

∂Λn would now be required to propagate the influence from Λn \B(e, r) to e; see

Figure 6.6(b). The proof of Lemma 6.3.6 is provided in Section 6.3.1.

The final ingredient in the proof of Theorem 6.3.5 is a LM estimate for Br with

r = Θ((log n)2). Such an estimate is readily provided by Theorem 5, with mixing

time that is poly-logarithmic in n.

Proof of Theorem 6.3.5. Let α > 0 be sufficiently large and let η ∈ Cα. By

Lemma 6.3.6, MSM holds for η and Br with r = Θ((log n)2) for some δ < n−3.

Observe also that every B(e, r) ∈ Br with boundary condition (1, η) or (0, η) is

a rectangle of side–length at most r = O((log n)2) with a realizable boundary

condition. Then by Theorem 5 (see also Remark 6.2.3), for every e ∈ E(Λn) we

have

maxtmix(B(e, r)0,η), tmix(B(e, r)1,η) ≤ (log n)C ,

for a suitable C > 0, yielding the desired LM estimate. The result then follows

from Theorem 6.0.5.

6.3.1 Moderate spatial mixing for Cα

We now prove Lemma 6.3.6. The proof involves showing that if ξ ∈ Cα, when

p < pc(q), the correlation between edges e, e′ ∈ E(Λn) near the boundary decays

exponentially in d(e, e′)/(α log n)—whereas SSM would entail a decay rate that is

exponential in just d(e, e′).

Proof of Lemma 6.3.6. Fix an edge e ∈ E(Λn) and for ease of notation let

197

B = B(e, r) ⊂ Λn and πη = πηΛn,p,q. Let (1, η) be the boundary condition induced

on B by η and the event Ec(B) = 1.

Lemma 6.2.5 implies that for every pair of configurations ω1, ω2 on Ec(B),

|πη( e = 1 | Ec(B) = ω1 )− πη( e = 1 | Ec(B) = ω2 )| ≤ π(1,η)(e η←→ ∂B \ ∂Λn

)(6.3.1)

where e η←→ ∂B \ ∂Λn denotes the event that there is a path from e to ∂B \ ∂Λn

taking into account the connections induced by η. Thus, it is sufficient to bound

the right-hand-side of (6.3.1).

There are three cases corresponding to the location of e in Λn. First, if

d(e, ∂Λn) > r, then B ∩ ∂Λn = ∅ and (1, η) is just the wired boundary con-

dition on B. In this case the right-hand-side of (6.3.1) is at most n−3 by the EDC

property; see (2.1.1).

The second and third cases correspond to whether B intersects one or two sides

of ∂Λn. For the second case, assume without loss of generality that B intersects

∂nΛn but not ∂wΛn or ∂eΛn. That is, d(e, ∂nΛn) ≤ r, but e is at distance

at least r from ∂wΛn and ∂eΛn. Let ∂wB, ∂sB, ∂eB be the west, south and east

boundaries of B, all of which are wired in ω. By a union bound

π(1,η)(e η←→ ∂B \ ∂Λn

)≤ π(1,η)

(e η←→ ∂wB

)+ π(1,η)

(e η←→ ∂eB

)+ π(1,η)

(e η←→ ∂sB

),

where e η←→ ∂wB denotes the event that there is a path from e to ∂wB in

B, taking into account those connections inherited from η (and ignoring the

connections induced by the wired configuration on Ec(B)). Define e η←→ ∂eB

198

and e η←→ ∂sB similarly.

The event e η←→ ∂sB implies that there exists a path of length at least r,

either from e or from ∂Λn to ∂sB. Therefore, the EDC property ((2.1.1)) implies

that for large enough n,

π(1,η)(e η←→ ∂sB

)≤ 1

3n3.

We bound next π(1,η)(e η←→ ∂wB). Let η0, . . . , ηd be the boundary com-

ponents of η. Since η is realizable, the planarity of Z2 implies that for every

i, j ∈ 0, . . . , d there are only three possibilities: L(ηi) ∩ L(ηj) = ∅, L(ηi) ⊂ L(ηj)

or L(ηj) ⊂ L(ηi). (Recall that L(ηi) ⊂ ∂Λn is the path of minimum length

that contains all the vertices in ηi.) Call ηi a maximal boundary component if

@j ∈ 0, . . . , d such that L(ηi) ⊂ L(ηj). The set of all maximal boundary compo-

nents defines a partition for ∂Λn. Since also η ∈ Cα, we deduce that there exists

a sequence of edges e0 = u0, v0, e1, . . . , ek = uk, vk in B ∩ ∂Λn such that 1)

(γ+α) log n ≥ d(ei, ei+1) ≥ γ log n for all i = 0, . . . , k−1, where γ is a large con-

stant we choose later and k ≥ c02(γ+α)

log n, and 2) the set Si = Jvi, ui+1K ⊂ B ∩ ∂Λn

is a disconnecting interval.

Let Ei be the event that Si is connected to Si+1 by a path of open edges in B.

Let et be the closest edge in the sequence e0, e1, . . . , ek to e and let Et = ∩ti=0Ei.

Since d(e, ∂wB) ≥ r, we also have t ≥ c08(γ+α)

log n. Then,

π(1,η)(e η←→ ∂wB

)≤ π(1,η)

(e η←→ ∂wB | Ect

)+ π(1,η)

(Et). (6.3.2)

If the event Ect occurs, then there exists i < t such that Si is not connected to Si+1

in B. This implies that there is a dual-path of length at least γ log n separating

199

Si from Si+1. Consequently, a path from e to ∂wB would require two vertices at

distance at least γ log n to be connected by a path of open edges in B. By the EDC

property and a union bound, this has probability at most 1/(9n3) for large enough

γ. Thus,

π(1,η)(e η←→ ∂wB | Ect

)≤ 1

9n3. (6.3.3)

We bound next π(1,η)(Et). Let ui, vi denote the endpoints of the edge ei, where

ui is to the left of vi for all i. For 1 ≤ i ≤ t, consider the rectangle Qi ⊂ B with

corners at vi−1, ui+3 and the other two corners on ∂sB. Then,

π(1,η)(Et) = π(1,η)(E0, . . . , Et) ≤ π(1,η)(E1, E5, E9 . . . , E`), (6.3.4)

where t − 4 < ` ≤ t. Now, let E ′i be the event that Si is connected to Si+1 by a

path completely contained in Qi. We have

π(1,η)(E1, E5, E9 . . . , E`) = π(1,η)(E ′1, E5, E9 . . . , E`) + π(1,η)(E1 ∩ (E ′1)c, E5, E9 . . . , E`)

≤ π(1,η)(E ′1, E5, E9 . . . , E`) +1

n4,

where the last inequality follows from the fact that for the event E1 ∩ (E ′1)c to occur

there have to be two vertices at distance at least γ log n connected by a path in

B; by the EDC property this only occurs with probability at most n−4 for large γ.

Iterating this procedure for E5, E9, . . . we get

π(1,η)(E1, E5, E9 . . . , E`) ≤ π(1,η)(E ′1, E ′5, E ′9 . . . , E ′`) +`

n4. (6.3.5)

200

Let Q =⋃(`−1)/4i=0 Q4i+1. Monotonicity implies that

π(1,η)(E ′1, E ′5, E ′9 . . . , E ′`) ≤ π1(E ′1, E ′5, E ′9 . . . , E ′` | Ec(Q) = 1)

=

`−14∏i=0

π1(E ′4i+1 | Ec(Q) = 1),

where for the last equality we use that the events E ′1, E ′5, . . . , E ′` are independent

under the wired boundary condition when also conditioning on Ec(Q) = 1 . We

claim that there exists a constant ρ ∈ (0, 1) (independent of n) such that for all

i = 0, . . . , `−14

π1(E ′4i+1 | Ec(Q) = 1) ≤ 1− ρ . (6.3.6)

To see this, let j = 4i+ 1 and note that for for large enough D > 0, by the EDC

property and a union bound imply that there is no connection between any pair of

vertices (u, v) with u ∈ Sj and v ∈ Sj+1 and d(u, v) ≥ D with probability Ω(1). At

the same time, by forcing an adjacent O(D) edges to be closed at a cost of e−O(D),

we see that with Ω(1) probability, in fact no other pairs (u, v) with d(u, v) ≤ D are

connected either. (6.3.6) holds. Thus,

π(1,η)(E ′1, E ′5, E ′9 . . . , E ′`) ≤ (1− ρ)`+3

4 ≤ (1− ρ)t−1

4 ≤ 1

9n3, (6.3.7)

where the last inequality holds for sufficiently large c0 since t ≥ c08(γ+α)

log n.

Putting (6.3.7), (6.3.5), (6.3.3), (6.3.4) and (6.3.2) together we get

π(1,η)(e η←→ ∂wB

)≤ 2

9n3+

`

n4≤ 1

3n3,

201

since ` = O(log n). Analogously, we get π(1,η)(e η←→ ∂eB) ≤ 13n3 , and thus

π(1,η)(e η←→ ∂B

)≤ 1

n3.

Finally for the third case, suppose without loss of generality that B intersects

∂nΛn and ∂wΛn, but not ∂eΛn or ∂sΛn. A union bound implies that

π(1,η)(e η←→ ∂B \ ∂Λ

)≤ π(1,η)

(e η←→ ∂sB

)+ π(1,η)

(e η←→ ∂eB

),

(6.3.8)

and each term in the right-hand side of (6.3.8) can bounded in the same way as

π(1,ξ)( e η←→ ∂wB ) in the second case; thus, the result follows.

6.4 A canonical paths bound with realizable FK

boundary conditions

As explained in §2.2.2, a classical and very fruitful bound using the canonical

paths method demonstrates that the Ising/Potts models on Λn,l have a mixing time

that is at most exponential in minn, l (for every β, q and boundary conditions):

see [28, 29, 53, 86]. However, FK boundary conditions on Λn,l can significantly

distort the cutwidth of the underlying graph encoding the interactions. Indeed

the lower bound in Theorem 7 shows that one could not possibly hope for such a

canonical paths bound in the presence of arbitrary FK boundary. Nonetheless, using

the nesting structure of disconnecting intervals of realizable boundary conditions,

we are able to prove a (weaker) bound that is exponential in (n ∨ l) log(n ∧ l) but

holds at every value of parameters p, q (including q < 1 as well as the critical p).

202

Unfortunately, this bound requires that the boundary conditions on one of the long

sides be either wired or free. The results in this section were not presented [8] but

are based on joint work with Blanca and Vigoda.

Theorem 6.4.1. Consider Λn,l = J0, nK × J0, lK for l ≤ n with realizable FK

boundary conditions (0, ξ) that are free on ∂sΛn,l and ξ on⋃i∈n,e,w ∂iΛn,l. Then for

every p ∈ (0, 1) and every q > 0, there exists c(p, q) > 0 such that

gap−1rc (Λ

(1,ξ)n,l ) . exp(c l log n) .

The same holds under (1, ξ) boundary conditions, denoting wired on ∂sΛn,l.

By duality, through Remark 2.2.1, it suffices to show the case of (0, ξ).

6.4.1 Tree structure of realizable boundary conditions

Recall the nesting properties of boundary bridges from Section 3.2 and of

disconnecting intervals from Section 6.1.2. In this section, we construct a tree Tξ of

rectangular subsets of Λn,l, corresponding to some realizable boundary condition ξ

on ∂nΛn,l, in order to encode the nested structure of the disconnecting intervals

of ξ. This will allow us to inductively bound the cutwidth of the graph Λ with

boundary conditions ξ.

Constructing the tree Tξ.

Recall that a rectangular subset of Λ = Λn,l is a subset of the form Ja1, a2K×J0, nK

for 0 ≤ a1 < a2 ≤ n; moreover, each rectangular subset is naturally identified

with the interval Ja1, a2K so that we call Ja1, a2K× J0, lK a disconnecting interval if

Ja1, a2K is one. Begin by identifying the root of the tree Tξ with Λ = J0, nK× J0, lK.

203

We define Tξ inductively from here. For any rectangular subset corresponding to a

vertex of Tξ, we describe how to assign its children rectangular subsets.

Consider a vertex w ∈ Tξ corresponding to rectangular subset Ja1, a2K× J0, lK.

If a2 = a1 + 1, then w will be a leaf of Tξ (and will have no children in Tξ). If

a2 > a1 + 1, then either w is the root of Tξ, or Ja1, a2K is a disconnecting interval

of wired-type, so that a1, a2 are in some boundary component Ci, and moreover

there will not exist any x ∈ (a1, a2) with (x, l) ∈ Ci. In this latter case, we define

the children of w in Tξ as follows:

1. Label the boundary components that are directly nested in Ci (i.e., not

counting those that are nested in a component Cj that is nested in Ci)

Ci1 , ..., Cik for some k. (Notice that singleton vertices with no bridges over

them before Ci count.)

2. Label the x–values of the vertices in⋃j Cij in increasing order by x1, ..., xK ,

with x1 = a1 + 1 and xK = a2 − 1, and set x0 = a1, xK+1 = a2.

3. The node w ∈ Tξ will have K + 1 children identified with the rectangular

subsets

Jxi, xi+1K× J0, lK for 0 ≤ i ≤ K .

Remark 6.4.2 (Well-definedness of Tξ). It should be clear from the above con-

struction that each of the children of w is either of the form xi+1 = xi + 1, or it is

a disconnecting interval of wired-type, with the property that xi and xi+1 are in

the same boundary component and there do not exist any x ∈ (xi, xi+1) with (x, l)

in that boundary component. As such, the above construction is well-defined.

204

Remark 6.4.3 (Properties of Tξ). With the well-definedness in hand, one can see

the following properties of Tξ. The tree Tξ has depth at most n, and there are

exactly n leaves corresponding to Ji, i+ 1K× J0, lK for all 0 ≤ i ≤ n− 1. Moreover,

for any node w ∈ Tξ, the disconnecting intervals of its children are strict subsets

of its disconnecting interval, and the union of the intervals corresponding to its

children will be exactly the interval corresponding to w.

Labeling the tree Tξ. For a given realizable boundary condition ξ on ∂nΛ, let Tξ

be the corresponding tree defined above. For a node w ∈ Tξ, we will let Rw denote

the corresponding rectangular subset. Moreover, if Rw = Ja, bK× J0, lK we recall its

width W (w) = W (Rw) = b− a, so that w is a leaf of Tξ if and only if W (w) = 1,

and the width of a child is strictly less than that of its parent.

It will also be important for us to have a particular ordering on the children of

a node w ∈ Tξ. We order the children so that the first child has the biggest width,

the next children are the neighboring rectangular regions to this first child, and

then continuing outwards.

Namely, for every w ∈ Tξ, label its children (w(i))i in any way that W (w(1)) ≥

W (w(j)) for every j, and such that⋃j≤k Rw(j) is a connected rectangular subset of

Λ for every k.

Remark 6.4.4. Notice crucially that any any such rectangular subset of the

form⋃j≤k Rw(j) can have at most two northern boundary components connect to

∂nΛ \⋃j≤k Rw(j) in ξ, and they must be through its two northern corner vertices.

205

6.4.2 Bounding the Cutwidth of Λn,l

Definition 6.4.5. For a fixed realizable boundary condition ξ on ∂Λn,l and any

enumeration of the edges in Λn,l as e1, ..., e|E(Λ)| identified with the bijection γ :

1, ..., |E(Λ)| → E(Λ), we can define

CWΛ,γ = maxk≤|E(Λ)|

|∂outγ(1), ..., γ(k)| ,

where for a set E(A) ⊂ E(Λ), |∂outA| is the number of vertices in ∂A that are

incident an edge in A and incident an edge in E(Λ) \ E(A) (where a boundary

vertex is incident to all the edges of E(Λ) adjacent to any vertex in its boundary

component in ξ). The cutwidth of Λ is given by

CWΛ = minγCWΛ,γ ,

where the minimum is over all bijections γ : 1, ..., |E(Λ)| → E(Λ).

For a rectangular subset R ⊂ Λ we define its cutwidth CWR with respect to

Λ by taking the minimum over bijections γ : 1, ..., |E(R)| → E(R) but defining

∂out as before, w.r.t. Λ.

Lemma 6.4.6. Consider Λ = Λn,l with boundary conditions (0, ξ) given by arbitrary

realizable boundary conditions ξ on ∂nΛ that is free on⋃i∈s,e,w ∂iΛ. For every

w ∈ Tξ, the rectangular subset Rw satisfies

CWRw ≤ CWRw(1)∨ (max

j≥2CWR

w(j)+ 2l + 2) ∨ (2l + 2) .

Corollary 6.4.7. Consider Λ with (0, ξ) boundary conditions given by realizable

boundary conditions ξ on ∂nΛ and free elsewhere. There exists C > 0 (independent

206

of ξ) such that

CWΛ ≤ Cl log n .

Proof. We will repeatedly apply Lemma 6.4.6 to bound CWRw along the tree Tξ.

We begin with the leaves of Tξ, each of which are single–column rectangular subsets.

For any leaf w ∈ Tξ, it has CWRw ≤ 2l + 2 (simply enumerate the edges in E(Rw)

in lexicographic order of their midpoints).

Then we can bound CWΛ by the maximum over all root–to–leaf paths of Tξ of

repeated application of Lemma 6.4.6 along those paths. More precisely, let a path

w = (w0, w1, ..., wf ) be a sequence of nodes in Tξ such that w0 is the root of Tξ, wf

is a leaf of Tξ and for every i, wi is a child of wi−1. Then applying Lemma 6.4.6 to

wf−1, ..., w0 and using that CWRwf≤ 2l + 2 and Rw0 = Λ, we see that

CWΛ ≤ (2l + 2) + maxw=(w0,...,wf )

∑1≤i≤f

(2l + 2) · 1wi 6= w(1)i−1 .

This is to say that when going down the path w there is no additive gain in the

cutwidth when the child of a node w is the child of largest width, and an additive

gain of at most 2l + 2 otherwise.

Now notice that for every node w ∈ Tξ and j ≥ 2, deterministically, W (w(j)) ≤12W (w). Combining this with the trivial fact that W (w) ≥ 1 for every w ∈ Tξ, we

see that along every root–to–leaf path w, at most log2 n of summands above are

nonzero. As a result we have

CWΛ ≤ (2l + 2)(1 + log2 n) ,

207

yielding the desired.

Proof of Lemma 6.4.6. Fix a w ∈ Tξ. We begin by defining an inductive algo-

rithm to choose

γ : 1, ..., |E(Rw)| → E(Rw)

given the enumerations γj : 1, ..., |E(Rw(j))| → E(Rw(j)) that attained the

cutwidth, i.e., CWRw(j)

= CWRw(j) ,γj . (Of course for any choice of γ, CWRw ≤

CWRw,γ.) We define the enumeration γ of the edges in E(Rw) as follows.

1. Recall the labeling of the children of w ∈ Tξ such that W (w(1)) ≥ W (w(j))

for every j and⋃j≤k Rw(j) is a connected rectangular subset.

2. For m ∈ 1, ..., |E(Rw(1))|, set γ(m) = γ1(m).

3. For every k ≥ 2, for every m ∈ |E(⋃j≤k−1Rw(j))| + 1, ..., |E(

⋃j≤k Rw(j))|

follow the ordering on edges in E(Rw(k)) given by γk, skipping over all edges

that are in E(⋃j≤k−1Rw(j)) and have thus already been enumerated.

We claim that this enumeration gives the desired inequality on CWRw,γ.

The idea here is that at any step k in the enumeration γ, the quantity

|∂out(⋃j≤k γ(j))| is at most the cutwidth of the child to which the most recently

enumerated edge belongs, plus the contribution from the children that have already

been fully enumerated. For the children that have already been enumerated, by

Remark 6.4.4, their contribution is at most 2l + 2. While enumerating the first

child (w(1)), this additional contribution is not present. We now make this intuition

precise.

First of all, if w is a leaf of Tξ, as noted earlier CWw ≤ 2l + 2 because Rw must

be single-column rectangular subset. Now assume that w is not a leaf of Tξ and

208

that it has children labeled w(1), w(2), ....

We consider the maximal cutwidth through the indices 1, ..., |E(Rw(1))| first,

then consider the maximal cutwidth in the rest of the enumeration separately.

Throughout the enumeration by indices m ∈ 1, ..., |E(Rw(1))|, by definition

of CWRw(1)

the maximal size of |∂out(⋃j≤m γ(j))| will be exactly the cutwidth

CWRw(1)

. Now, let’s consider the largest |∂out(⋃j≤m γ(j))| gets through indices

m ∈ |E(⋃j≤k−1Rw(j))|+ 1, ..., |E(

⋃j≤k Rw(j))| for any k ≥ 2. For all such m, we

have

|∂out(⋃j≤m

γ(j))| ≤ CWRw(k)

+ |∂out(⋃

j≤k−1

Rw(j))| ,

and therefore,

CWRw,γ ≤ CWRw(1)∨(

maxk≥2

CTWRw(k)

+∣∣∣∂out( ⋃

j≤k−1

Rw(k)

)∣∣∣) .By Remark 6.4.4, the contribution of

⋃j≤k−1Rw(j) is at most 2l+ 2 for every k ≥ 2,

concluding the proof.

6.4.3 From cutwidth to spectral gap

We now use the classical canonical paths approach used for short range spin-

systems to translate Corollary 6.4.7 into a proof of Theorem 6.4.1. Recall the

canonical paths bound on the inverse gap given by Theorem 2.2.9.

Proof of Theorem 6.4.1. We first move from boundary conditions that are ar-

bitrary realizable boundary conditions on⋃i∈n,e,w ∂iΛ to those that are arbitrary

realizable on ∂nΛ and free on ∂eΛ∪ ∂wΛ, by (2.2.9), absorbing a multiplicative cost

209

of nl · ecl in the spectral gap.

Now fix any realizable boundary condition ξ on ∂nΛ, let γ be an enumeration

of E(Λ) satisfying CWΛ,γ ≤ Cl log n (whose existence is guaranteed by Corollary

6.4.7) and label the edges E(Λ) according to γ as e1, ..., e|E(Λ)|. For any two FK

configurations ω, ω′ ∈ Ωrc on Λ, we define a path ζ(ω, ω′) in Ωrc as follows: let

el1 , ..., elk ∈ E(Λ) be the sequence of edges on which ω(eli) 6= ω′(eli), labeled in

the order dictated by γ. Then for every i, the i’th edge of ζ will be between the

configuration

ηi(ej) =

ω′(ej) for j ≤ li − 1

ω(ej) for j ≥ li

,

and its neighbor in Ω, η′i which instead has η′(eli) = ω′(eli). At the same time, we

can define

ηi(ej) =

ω(ej) for j ≤ li − 1

ω′(ej) for j ≥ li

.

Then, using the definition of π, it is clear that for every i,

πξ(ω)πξ(ω′) ≤ πξ(ηi)πξ(ηi)q

2·CWΛ,γ

as (recalling that k(ω; ξ) is the number of connected components using boundary

conditions ξ in configuration ω)

|k(ηi; ξ) + k(ηi; ξ)− k(ω; ξ)− k(ω′; ξ)| ≤|∂outel1 , ..., eli−1|+ |∂outeli , ..., e|E(Λ)|| ,

210

which is in turn at most 2|∂oute1, ..., eli−1|. By construction, for every i, the map

(ω, ω′) 7→ (ηi, ηi) is injective. Moreover, the probability of FK dynamics making

any transition in Ωrc is bounded below by some ρ(p, q) > 0. Plugging these in to

Theorem 8.0.6, we obtain for some C > 0,

gap−1rc ≤ max

η,η′∈Ω

1

ρπ(η)

∑ω,ω′:(η,η′)∈ζω,ω′

nlπ(η)π(η)q2·CWΛ,γ

≤ ρ−1 · nl · qCl logn

which for all l, n sufficiently large satisfies the desired for a different C.

6.5 Slow mixing under worst-case boundary con-

ditions

In this section we show that there are (non-realizable) boundary conditions for

the graph (Λn, E(Λn)) for which the FK-dynamics requires exponentially many

steps to converge to stationarity. In particular, we prove Theorem 7 from the

introduction.

Theorem 7 is a corollary of a more general theorem we establish. This general

theorem enables the transferring of mixing time lower bounds for the FK-dynamics

on arbitrary graphs to mixing time lower bounds for the FK-dynamics on Λn, for

suitably chosen boundary conditions. The high level idea is that any graph G with

fewer than n4

edges can be “embedded” into a subset L of the boundary ∂nΛn of

Λn as a boundary condition we shall denote ξ(G). When p is sufficiently small, the

effect of the configuration on Λn \ L becomes negligible, and so the mixing time of

211

the FK-dynamics on Λn with boundary condition ξ(G) is primarily dictated by its

restriction to the embedded graph G.

We show first how to embed any graph G = (VG, EG) into a subset L ⊂ ∂nΛn.

For m ≤ bn/4c, let

L = L(m) = J4i, 4i+ 1K : i = 0, ...,m− 1 × n ⊂ ∂nΛn

with edge set E(L) consisting of all edges in E(Λn) connecting vertices in L.

Definition 6.5.1. Let G = (VG, EG) be a graph with |EG| = m for m ≤ bn/4c

and let L be as above. We say a function φ : L→ VG is an embedding of G into

(L,E(L)) if for every u, v ∈ EG there exists a unique pair x ∈ φ−1(u) ⊆ L and

y ∈ φ−1(v) ⊆ L, where φ−1(u) and φ−1(v) denotes the pre-image sets for u and v

respectively, such that x, y ∈ E(Λn).

Notice that every graph G on m ≤ bn/4c edges can be embedded into L by

identifying each edge in EG with an edge in E(L).

Fact 6.5.2. For every graph G = (VG, EG) with m ≤ bn4c edges, there exists an

embedding of G into (L,E(L)).

Now let ξ(G) be the boundary condition on ∂Λn defined by the partition:

v : v ∈ ∂Λn \ L ∪ φ−1(v) : v ∈ VG .

In words, ξ(G) is the boundary condition that is free everywhere except in the

vertices of L and where all the vertices in L that are mapped by φ to the same

vertex of G are wired in ξ(G). We are now ready to state our main comparison

result from which Theorem 7 follows straightforwardly.

212

Theorem 6.5.3. Let G = (VG, EG) be a graph and suppose there exist q > 2 and

p = λ|EG|−α with λ > 0, α > 1/3 such that gap(G) ≤ exp(−Ω(|VG|)). Then, as

long as n ≥ 4|EG| ≥ εn for some ε > 0, with the same choice of p and q, we have

gap(Λξ(G)n ) ≤ e−Ω(|VG|) .

Proof of Theorem 7. It was established in [44] that for every q > 2 and every `

sufficiently large, there exists an interval (λs(q), λS(q)) such that if p′ = λ′/` with

λ′ ∈ (λs(q), λS(q)), then the spectral gap at parameters (p′, q) satisfies

gap(K`) ≤ exp(−Ω(`)) ,

where K` denotes the complete graph on ` vertices. Therefore, for a fixed p = λn−α

there exists a choice of ` = Θ(nα) such that at parameters (p, q), gap(K`) ≤

exp(−Ω(`)). Since the number of edges in K` is Θ(n2α), the result follows from

Theorem 6.5.3 and (2.2.3).

6.5.1 Main comparison inequality: proof of Theorem 6.5.3

We now turn to the proof of Theorem 6.5.3. A standard tool for bounding

spectral gaps is construction of bottleneck sets with small conductance; see Sec-

tion 2.2.1. It will be easier to do so for the following modified heat-bath (MHB)

dynamics, allowing us to isolate moves on E(L), where we have embedded G, from

those in Ec(L) = E(Λn) \ E(L).

Definition 6.5.4. Given an FK configuration Xt, one step of the MHB chain is

given by:

213

1. Pick e ∈ E(Λn) uniformly at random;

2. If both endpoints of e lie in L, then perform a heat-bath update on e. That is,

replace the configuration in e with a sample from πξΛn,p,q(· | Xt(E(Λn) \ e));

3. Otherwise, replace the configuration in E(Λn) \ E(L) with a sample from

πξΛn,p,q(· | Xt(E(L))).

The MHB chain is clearly reversible w.r.t. πξΛn,p,q.

Let gapmhb(Λξn) denote the spectral gap of the MHB dynamics on Λn with

boundary condition ξ and parameters p and q. The following comparison inequality

allows us to focus on finding upper bounds for the spectral gap of the MHB

dynamics; its proof is deferred to Section 6.5.2.

Lemma 6.5.5. For all p ∈ (0, 1), q > 0, n ∈ N and boundary condition ξ for Λn,

we have

gap(Λξn) ≤ gapmhb(Λξ

n) .

With this in hand, we are now ready to prove Theorem 6.5.3.

Proof of Theorem 6.5.3. Recall from (2.2.15) that since, by assumption, the

FK-dynamics on G has gaprc(G) ≤ exp(−Ω(|VG|)), there must exist S? ⊂ ΩG (the

set of FK configurations on G) with πG(S?) ≤ 12

such that

Φ(S?) =QG(S?, S

c?)

πG(S?)≤ e−Ω(|VG|) . (6.5.1)

Here QG is the edge measure (2.2.14) of the FK-dynamics on G and πG = πG,p,q

denotes the random-cluster measure on G. We will construct from this set S?, a

214

set A? ⊂ Ω, such that :

Φ(A?) =Qmhb(A?, A

c?)

πξ(G)(A?)≤ e−Ω(|VG|) , and

Φ(Ac?) =Qmhb(Ac?, A?)

πξ(G)(Ac?)≤ e−Ω(|VG|) , (6.5.2)

where Qmhb denotes the edge measure (2.2.13) of the MHB dynamics on Λξ(G)n

and πξ(G) = πξ(G)Λn,p,q

. This implies Theorem 6.5.3 by combining it with (2.2.3)

and (2.2.15).

Let ξ1, . . . , ξk be the partition of L induced by ξ(G). For an FK configuration

ω on Ec(L), we say that ξiω←→ ξj if there is an open path in ω from a vertex in ξi

to a vertex in ξj. Let Sξ(G)(ω) be the set

Sξ(G)(ω) = ξi ∈ ξ(G) : ξiω←→ ξj for some j 6= i, j ∈ 1, . . . , k ,

i.e., those ξi that are connected to some ξj in ω. For M ≥ 0, let

Rξ(G)(M) = ω ∈ 0, 1Ec(L) : |Sξ(G)(ω)| ≤M .

In words, Rξ(G)(M) is the set of FK configurations on E(Λn) \ E(L) that connect

at most M elements of the partition ξ1, . . . , ξk of the vertex set L.

Observe that any configuration θ on E(L) corresponds to a configuration on EG.

Namely, if θ(u, v) = 1, then the edge φ(u), φ(v) is open in the configuration on

G, where φ is the embedding of G into L. With a slight abuse of notation we may

use θ also for the corresponding configuration on ΩG. With this convention, let

AM = ω ∈ Ω : ω(E(L)) ∈ S? , ω(Ec(L)) ∈ Rξ(G)(M) . (6.5.3)

215

We show that if M = δ|VG| for some δ > 0 sufficiently small, and n is taken to

be large enough, we can take A? = AM . This will follow from the following two

claims.

Claim 6.5.6. The following are true of S? and AM defined above:

(i) πξ(G)(AM) ≥ q−M(1− e−Ω(M))πG(S?) ;

(ii) πξ(G)(AcM) ≥ e−O(M) .

Claim 6.5.7. The modified heat-bath dynamics satisfies

Qmhb(AM , AcM) ≤ πξ(G)(AM)e−Ω(M logn) +

q2M+1

pQG(S?, S

c?) .

Dividing the bound from Claim 6.5.7 by πξ(G)(AM) and using the bounds from

Claim 6.5.6, we see that

Qmhb(AM , AcM)

πξ(G)(AM)≤ e−Ω(M logn) +

2q3M+1

p

QG(S?, Sc?)

πG(S?)≤ e−Ω(M logn) + eO(M)e−Ω(|VG|) .

for sufficiently large M , where the last inequality follows from (6.5.1) and the facts

that M = δ|VG| and p ≥ λ|VG|−2α. Similarly, we get

Qmhb(AM , AcM)

πξ(G)(AcM)≤ e−Ω(M logn) + eO(M)e−Ω(|VG|) .

Then, since M = δ|VG|, for some δ > 0 sufficiently small we obtain (6.5.2).

6.5.2 Proof of auxiliary facts

In this section we provide the proofs of Lemma 6.5.5, and Claims 6.5.6 and 6.5.7.

216

Proof of Lemma 6.5.5. For any B ⊂ E(Λn), let PB be the transition matrix

corresponding to a heat-bath update on the entire set B. For e ∈ E(Λn), we use

Pe for Pe. Let Phb and Pmhb be the transition matrices for the FK-dynamics and

the MHB dynamics on Λξn, respectively. Let A = E(Λn) \ E(L). Then,

Pmhb =1

|E(Λn)|

( ∑e∈E(L)

Pe +∑e∈A

PA

).

For ease of notation, set π = πξΛn,p,q. Then, for any f, g ∈ R|Ω|, where Ω denotes

the set of FK configurations on Λn, let

〈f, g〉π =∑ω∈Ω

f(ω)g(ω)π(ω) .

If we endow R|Ω| with the inner product 〈·, ·〉π, we obtain a Hilbert space denoted

L2(π) = (R|Ω|, 〈·, ·〉π). Recall, that if a matrix P is reversible w.r.t. π, it defines

a self-adjoint operator from L2(π) to L2(π) via matrix vector multiplication, and

thus 〈f, Pg〉π = 〈P ∗f, g〉π = 〈Pf, g〉π.

Now, the matrix Pmhb is positive semidefinite, since it is an average of pos-

itive semidefinite matrices. Hence, it is a standard fact (see, e.g., [60]) that if

〈f, Pmhbf〉π ≤ 〈f, Phbf〉π for all f ∈ R|Ω|, then gapmhb(Λξn) ≥ gaprc(Λξ

n). To show

this, note that for e ∈ A, we have PA = PePAPe. Thus, for all f ∈ R|Ω|,

〈f, PAf〉π = 〈f, PePAPef〉π = 〈P ∗e f, PAPef〉π = 〈Pef, PAPef〉π

≤ 〈Pef, Pef〉π = 〈f, Pef〉π ,

where we used that Pe = P ∗e , Pe = P 2e and that 〈f, Pf〉π ≤ 〈f, f〉π for every

217

f ∈ R|Ω| and every matrix P reversible w.r.t. π. Then,

〈f, Pmhbf〉π ≤1

|E(Λn)|∑

e∈E(Λn)

〈f, Pef〉π = 〈f, Phbf〉π,

and the result follows.

Recall the notation of Lemma 6.5.3. In order to compare the marginal distribu-

tion in L to πG, we need to bound the number of connections in Ec(L) between

different boundary components of ξ(G) restricted to L: this will show that typical

FK configurations on Ec(L) do not have much influence on the connectivities

amongst L. This bound follows from the fact that p = O(n−α) for some α > 1/3

and the (approximate) independence of connections between ξi and ξj. For the

reminder of this section we set ξ = ξ(G) for ease of notation. Claims 6.5.6–6.5.7

will then be seen as consequences of the following lemma.

Lemma 6.5.8. Let p = λn−α with λ > 0 and α > 1/3. Let ξ be any boundary

condition on ∂nΛn and let η be any FK configuration on E(L). Then, when q ≥ 1,

for every M ≥ 1,

πξΛn(Rξ(M) | η

)≥ 1− exp

[−Ω

(M log[Mn3α−1]

)].

Proof. Let Y be the random variable for the number of vertices of L connected to

at least one other vertex of L in an FK configuration on Ec(L) sampled from the

distribution πξ,η(·) = πξΛn,p,q(· | η). It is sufficient to show that

πξ,η(Y ≥M) ≤ exp(−Ω

(M log[Mn3α−1]

)).

218

By classical comparison inequalities (see e.g., [47]), when q ≥ 1, no matter the

boundary conditions ξ, η, the random-cluster measure on Ec(L) is stochastically

dominated by the independent bond percolation distribution on Ec(L) with the

same parameter p, which we denote by ν = νΛn,p. (Recall that ν is the distribution

on Ω that results from adding every edge in E(Λn) independently with probability

p.) Hence, if X is defined as Y but for ν on Ec(L), we get

πξ,η(Y ≥M) ≤ ν(X ≥M).

Consider the subgraph Λ = (Λn, Ec(L)). Let Z be the total number of vertices

in L that are connected in a configuration sampled from ν to another vertex at

distance 3 in Λ. Then, since the distance between any two vertices in L in Λ is at

least 3, we see that X ≤ Z and

ν(X ≥M) ≤ ν(Z ≥M).

Thus, it suffices to establish a tail bound for Z. Enumerate the vertices of L

as v1, . . . , v2m and let Zj be the indicator random variable for the event that vj

is connected to another vertex at distance 3 in Λ. For r = 0, 1, 2, 3, split up

Z =∑

r Zr, where

Zr =∑i≥0

Z4i+r+1 .

We claim that there is some suitable c > 0 such that under ν, each Zr is stochastically

dominated by the binomial random variable S ∼ Bin(dn/4e, cn−3α). This is because

the random variables Z4i+r+1 are jointly dominated by independent Bernoulli

random variables, Ber(cn−3α), for a suitable c > 0, since for every k the events

219

Zk = 1 and Zk+4 = 1 depend on disjoint sets of edges. The fact that the

success probability of each one is at most cn−3α follows from the fact that there are

at most 16 choices of three adjacent edges from a vertex in ∂nΛn, and p = λn−α.

Hence, by Chernoff–Hoeffding inequality, for every δ > 0,

ν(Zr ≥ Eν [S] + δdn/4e) ≤ exp[−n

4D(cn−3α + δ ‖ cn−3α

)].

where D(a‖b) is the relative entropy between the Bernoulli random variables Ber(a)

and Ber(b):

D(a‖b) = a log(a/b) + (1− a) log((1− a)/(1− b)) .

Since Eν [S] = O(n1−3α), Var(S) = O(n1−3α), α > 13

and M ≥ 1, it follows that for

every M ≥ 1,

ν(Zr ≥M/4) ≤ e−Ω(M log[Mn3α−1]) .

We have Z = Z1 + Z2 + Z3 + Z4, and so a union bound implies the matching bound

for ν(Z ≥M).

Now recall the definitions of the set S? and AM from (6.5.3).

Proof of Claim 6.5.6. For part (i), observe that if ω is sampled from πξ, then

πξ(AM) = πξ(ω(L) ∈ S? | ω(Ec(L)) ∈ Rξ(M)

)πξ(ω(Ec(L)) ∈ Rξ(M)

).

By Lemma 6.5.8,

πξ(ω(Ec(L)) ∈ Rξ(M)) ≥ 1− e−Ω(M) .

220

Moreover, since

πξ(ω(L) ∈ S? | ω(Ec(L) = 0)) = πG(S?) ,

it follows from Eq. (2.2.10) that

πξ(ω(L) ∈ S? | ω(Ec(L)) ∈ Rξ(M)) ≥ q−MπG(S?) ,

and thus,

πξ(AM) ≥ q−M(1− e−Ω(M))πG(S?).

Similarly for part (ii), we have

πξ(AcM) ≥ πξ(ω(L) 6∈ S? | ω(Ec(L)) ∈ Rξ(M)

)πξ(ω(Ec(L)) ∈ Rξ(M))

≥ q−M(1− e−Ω(M))πG(Sc?)

which is at least e−O(M) since πG(S?) ≤ 12.

Proof of Claim 6.5.7. Let Pmhb be the transition matrix for the MHB dynamics

and for ease of notation set B = Ec(L). We have

Qmhb(AM , AcM) ≤

∑ω∈AM

∑ω′∈Ω:

ω′(B)/∈Rξ(M)

πξ(ω)Pmhb(ω, ω′)

+∑ω∈AM

∑ω′∈Ω:

ω′(L)/∈S?

πξ(ω)Pmhb(ω, ω′) . (6.5.4)

For the first term in (6.5.4), observe by definition of MHB dynamics, for every

221

ω ∈ AM

∑ω′∈Ω:

ω′(B)/∈Rξ(M)

Pmhb(ω, ω′) ≤∑ω′∈Ω:

ω′(B)/∈Rξ(M)

πξ(ω′(B) | ω(L)

)≤ e−Ω(M logn),

where the last inequality follows from Lemma 6.5.8. Hence,

∑ω∈AM

∑ω′∈Ω:

ω′(B)/∈Rξ(M)

πξ(ω)Pmhb(ω, ω′) ≤ πξ(AM)e−Ω(M logn).

For the second term in (6.5.4), observe that ω 6= ω′ and that ω and ω′ can differ

in at most one edge e; otherwise Pmhb(ω, ω′) = 0. Thus, setting

p+(q) = max

p,

q(1− p)q(1− p) + p

, and p−(q) = min

1− p, p

q(1− p) + p

,

we obtain

Pmhb(ω, ω′) =1

|E(Λn)|π(ω′(e) | ω(E(Λn) \ e)

)≤ p+(q)|EG|p−(q)|E(Λn)|

PG(ω(L), ω′(L)) .

Then, since |EG| ≤ |E(Λn)| and p+(q)/p−(q) ≤ q/p,

∑ω∈AM

∑ω′∈Ω:

ω′(L)/∈S?

πξ(ω)Pmhb(ω, ω′)

≤ q

p

∑θ∈Rξ(M)

πξ(θ)∑ω1∈S?

∑ω2 6∈S?

πξ(ω1 | θ)PG(ω1, ω2)

≤ q

pπξ(Rξ(M))

∑ω1∈S?

∑ω2 6∈S?

maxθ∈Rξ(M)

πξ(ω1 | θ)PG(ω1, ω2),

222

where ω1, ω2 are FK configurations on E(L) and θ is an FK configuration on B.

Eq. (2.2.10) implies

maxθ∈Rξ(M)

πξ(ω1 | θ) ≤ q2MπG(ω1),

and so

∑ω∈AM

∑ω′∈Ω:

ω′(L)/∈S?

πξ(ω)Pmhb(ω, ω′) ≤ q2M+1

p

∑ω1∈S?

∑ω2 6∈S?

πG(ω1)PG(ω1, ω2)

=q2M+1

pQG(S?, S

c?).

Combining these two bounds, we get

Qmhb(AM , AcM) ≤ πξ(AM)e−Ω(M logn) +

q2M+1

pQG(S?, S

c?) .

223

Chapter 7

Boundary conditions in

coexistence regions I: free and

monochromatic boundary

In this chapter and Chapter 8, we analyze the sensitivity to boundary conditions

in coexistence regions of the FK model, namely at q > 4 and p = pc(q). Whereas

at continuous phase transition points, our mixing upper bounds held for both the

torus and typical FK boundary conditions, and are believed to be independent of

(realizable) FK boundary conditions, in the presence of phase coexistence, natural

boundary conditions like combinations of free, wired, and periodic can lead to

drastically different temporal mixing. This phenomenon can be seen in analogy to

the Ising Glauber dynamics at low temperatures, where the sensitivity to boundary

conditions is very well-studied [11, 52, 63, 66, 70]. We begin with the simplest case

of this, considering the free and wired boundary conditions (corresponding to free

and monochromatic Potts boundary conditions) at large q and p = pc(q).

224

We prove the sub-exponential mixing for the FK Glauber dynamics using

censoring inequalities (Theorem 2.2.7). A bound on the FK dynamics then implies

the analogous bounds for Chayes–Machta and Swendsen–Wang via Theorem 1.2.2.

The monotonicity requirement of the censoring prevents us from carrying this out

in the setting of the Potts Glauber dynamics, though we expect similar pictures

to hold there. As in [70], we will work directly with distributions over boundary

conditions induced by infinite-volume measures; this is more delicate in the setting

of the FK model where boundary interactions are no longer nearest-neighbor. We

now formally define these distributions. Throughout the chapter, let q > 4 and

p = pc(q) so that π1Z2,p,q 6= π0

Z2,p,q and drop the p, q from the notation.

Definition 7.0.1 (wired/free-at-infinity b.c.). In order to sample a boundary

condition on ∆ ⊂ ∂Λ from π0Z2 or π1

Z2 we sample the infinite-volume configuration

on E(Z2) − E(Λ) and then identify the induced boundary condition with the

partition of ∆ that is induced by that configuration. Call the distribution over

boundary conditions induced by π1Z2(ωE(Z2)−E(Λ) ∈ ·) wired-at-infinity and that

induced by π0Z2(ωE(Z2)−E(Λ) ∈ ·) free-at-infinity.

We say that a distribution P on boundary conditions on ∆ dominates P′

(denoted P P′) if ξ′ ∼ P′ is a stochastically finer partition than ξ ∼ P. Moreover,

if boundary conditions are given by different distributions on different subsets of

∂Λn,m, then define the overall distribution on boundary conditions by sampling

the boundary partitions independently on each of the boundary subsets (with no

connections between the subsets). We say such a boundary condition piecewise

dominates wired-at-infinity and is dominated by free-at-infinity.

A particular example of random boundary conditions that will be useful later

on is the following: if R ⊂ D are two concentric rectangles, then the boundary

225

conditions induced by the connections from ωE(D)−E(R), where ω is sampled from

π1D, dominate the wired-at-infinity boundary conditions.

We prove the following proposition, from which Theorem 8 follows easily.

Proposition 7.0.2. Let q be sufficiently large and consider Glauber dynamics for

the critical FK model on Λ = Λn,n. Let P be a distribution over boundary conditions

on ∂Λ such that P π0Z2 or P π1

Z2 and let E be its corresponding expectation.

Then for every ε > 0, there exists c(ε, q) > 0 such that for t? = exp(cn3ε),

maxω0∈0,1

E[‖P t?(ω0, ·)− πξΛ‖tv

]. e−cn

2ε

,

where ξ ∼ P. In particular, P(tmix & t?) . exp(−cn2ε).

Proof ideas

In order to prove Proposition 7.0.2, we appeal to the Peres–Winkler censoring

inequalities [76] for monotone spin systems, a crucial part of the analysis in [70]

(then later in [63]) of the low-temperature Ising model under “plus” boundary, a

class of boundary conditions that dominate the plus phase. A major issue when

attempting to adapt this approach to the critical FK model at sufficiently large q

with free-at-infinity boundary conditions is that the typical free-at-infinity boundary

conditions still have many boundary connections, inducing problematic long-range

interactions along the boundary (see Fig. 1.5), which, as in other previous chapters,

prevent coupling beyond interfaces.

To remedy this, at every step of the analysis we modify the boundary conditions

to all-free on appropriate segments of length no(1) (this modification can only affect

the mixing time by an affordable factor of exp(no(1))). With high probability, by the

226

exponential decay of correlations under π0Z2 , no boundary connections circumvent

the interface past the modified boundary. Refined large deviation estimates on

fluctuations of FK interfaces then allow us to control the influence of other long-

range boundary interactions (see Figure 7.3) and to couple different configurations

beyond distance n1/2+o(1) (the length scale that captures the normal fluctuations of

the interface), yielding mixing estimates on n× n1/2+o(1) boxes, the basic building

block of the proof. These precise large-deviation estimates are only attainable using

cluster expansion, which forces us to only work with sufficiently large q (as opposed

to all q > 4 where the phase transition is discontinuous); we develop these necessary

equilibrium estimates using cluster expansion in the following Section 7.1, before

proving the main result in Section 7.2.

7.1 Cluster expansion and equilibrium estimates

7.1.1 Surface tension

The order-disorder surface tension will play a large role in both upper and

lower bounds studying the effect of boundary conditions on mixing in the phase

coexistence regime. In the low-temperature regime, the metastable phases are the

q ordered ones and there is a positive order-order surface tension (see, e.g., the

q = 2 case); this leads to sensitivity of mixing times to boundary conditions in

Ising/Potts Glauber dynamics (cf., e.g., [66, 70]), but not in FK/Swendsen–Wang

dynamics (as these are symmetric w.r.t. the q ordered phases). At the critical point,

when q > 4, the disordered phase is also metastable, and should induce similar

sensitivity to boundary conditions in FK Glauber and Swendsen–Wang dynamics.

Let Sn = J0, nK × J−∞,∞K and let (1, 0, φ) FK boundary conditions on ∂Sn

227

denote those that are wired on ∂Sn ∩ (x, y) ∈ Z2 : y ≥ x tanφ and free elsewhere

on ∂Sn. We will always be taking φ ∈ (−π/2, π/2).

Definition 7.1.1. The order-disorder surface tension on Sn in direction φ is given

by

τ1,0(φ) = limn→∞

cosφ

βcnlog

[ Z1,0,φSn

(Z1SnZ

0Sn)1/2

]

(whenever this limit exists) where ZξSn denotes the FK partition function on Sn

with boundary conditions ξ on ∂Sn.

It was proved first in [57] that for q sufficiently large, at p = pc(q) and φ 6= ±π2,

the surface tension τ1,0(φ) exists and satisfies τ1,0(φ) > 0.

7.1.2 Cluster expansion

We first introduce the cluster expansion framework used to prove that the FK

order-disorder surface tension τ1,0(φ) as defined in Definition 7.1.1 is positive for all

large q at p = pc(q). We skip many of the details here as understanding them is not

necessary to the equilibrium estimates we require. This approach was extensively

developed in [30] for the low-temperature Ising model when β βc, and then

extended to the critical Potts/FK model for large q (at the discontinuous phase

transition point) in [72] where one can find more details on the below. Though these

cluster expansion techniques only go through at p = pc(q) when q is sufficiently

large, all of the interface estimates are expected to hold at pc whenever τ1,0(φ) > 0,

so in particular, for every q > 4.

Notation. In what follows, two (primal) edges e, f are adjacent if they share a

vertex. Two primal edges e, f are co-adjacent if e? is adjacent to f ?. Connectedness

228

and co-connectedness are defined naturally with respect to adjacency and co-

adjacency. For an edge subset A ⊂ E(Z2), the (edge)-boundary of A is the set of

all edges in A that are co-adjacent to E(Z2) \ A. The co-boundary of A is the set

of edges in E(Z2) \ A that are adjacent to A.

Definition 7.1.2. Let D be a connected subgraph of Z2 and let a, b be two marked

boundary points on ∂D. Consider Dobrushin boundary conditions that are wired

on the clockwise segment (a, b) and free on (b, a) (where if D is infinite, simply

connected, we define these boundary arcs in the natural way). Then for an FK

realization ω on D, the primal-FK interface (or simply the FK interface), I = I(ω),

is defined as follows:

1. Consider the dual-component of the boundary arc (b, a) in ω, and consider

its co-boundary (a set of closed dual-edges in ω).

2. This co-boundary has a unique co-connected component, call it C?(ω), that

is incident to the boundary arc (a, b). The primal-FK interface I is the set of

all primal edges that are dual to edges in C?(ω).

(Notice that this interface is not necessarily a simple path, but it is connected,

and will have no cycles.) Define the dual-FK interface analogously as follows:

consider the connected component of open edges touching the boundary arc (a, b)

and consider its co-boundary (a set of closed primal edges). This co-boundary

has a unique co-connected component that is incident the boundary arc (b, a);

the set of dual-edges that are dual to edges in this co-connected component will

be the dual-FK interface. For more general boundary conditions, there will be a

compatible collection of interfaces between all boundary segments that are wired.

Recall that we defined the infinite strip Sn = J0, nK× J−∞,∞K.

229

Definition 7.1.3. For any angle φ ∈ (−π2, π

2), an edge-cluster weight function

Φ(C, I) is any real-valued function with first argument that is a connected set of

edges in Sn and second argument that is a possible realization of an FK interface

with Dobrushin boundary conditions that are wired on the clockwise arc between

(0, 0) to (n, bn tanφc), such that for some λ > 0 and every C, I,

1. Φ(C, I) = 0 when C ∩ I = ∅ ,

2. Φ(C, I1) = Φ(C, I2) when ΠC ∩ I1 = ΠC ∩ I2 ,

3. Φ(C, I) = Φ(C + (0, s), I1) when I1 ∩ ΠC = (0, s) + I ∩ ΠC ,

4. |Φ(C, I)| ≤ exp(−λ`(C)) ,

where

ΠC = (x, y) ∈ R2 : ∃y′ s.t. (x, y′) ∈ C ,

and `(C) is the minimum number of edges in a connected subset of E(Z2) that

contains all the boundary edges of C. More generally, for an edge-cluster weight

function Φ and a domain Vn with Dobrushin boundary conditions between (0, 0)

and (n, nbtanφc), its partition function is given by

ZΦ = ZΦ(Vn, φ) =∑I

λ|I|+∑C:C∩I=∅ Φ(C,I) , (7.1.1)

where the sum runs over all possible FK order-disorder interfaces in Vn.

There exists a unique edge-cluster weight function Φo/d—called the FK order-

disorder weight function—such that the probability that the FK interface in Sn

under the (1, 0, φ) boundary condition is I is given by

π(1,0,φ)Sn (I) = Z−1

Φo/dλ|I|+

∑C:C∩I6=∅ Φo/d(C,I) .

230

The FK order-disorder weight function Φo/d is made explicit in [72, Proposition 5].

By adapting the methods of [30] to the FK cluster expansion, the following large

deviation estimate on order-disorder interface fluctuations was obtained in [72].

Proposition 7.1.4 ([72, Proposition 5]). Consider the critical FK model on Sn and

fix a δ > 0. There exist some q0 and c(δ) > 0 such that for all φ ∈ [−π2

+ δ, π2− δ]

and every q ≥ q0, every h ≥ 1 and n ≥ 1,

π1,0,φSn

(I 6⊂ (x, y) : y ∈ [x tanφ− h, x tanφ+ h]

). n2 exp(−ch2/n) .

The following is a finer result, that reformulates results of [30] in the FK setting.

Define the cigar-shaped region Uκ,d,φ for every d > 0 and κ > 0 by

Uκ,d,φ = Sn ∩

(x, y) ∈ Z2 : |y − x tanφ| ≤ d∣∣∣x(n−x)

n

∣∣∣ 12

+κ, (7.1.2)

Proposition 7.1.5. Consider the critical FK model on a domain Vn ⊃ Uκ,d,φ, let

Φo/d be the FK order-disorder weight function, and let Φ be any function satisfying

Φ(C, I) = Φo/d(C, I) when C ⊂ Uκ,d,φ , |Φ(C, I)| ≤ exp(−λ`(C)) ∀C, I ,

(e.g., Φ = Φo/d1C ⊂ Uκ,d,φ). There exist q0 > 0 and f(κ) = O(κ−1) such that for

all q ≥ q0, all φ ∈ [−π2

+ δ, π2− δ], there exists C(q, δ) > 0, such that

| logZΦ(Vn, φ)− logZΦo/d(Sn, φ)| ≤ C(log n)f(κ) . (7.1.3)

231

Moreover, the order-disorder surface tension τ1,0(φ) (given in Def. 7.1.1) satisfies

| logZΦ(Vn, φ)− n log(1 +√q)(cosφ)−1τ1,0(φ)| ≤ C(log n)f(κ) . (7.1.4)

7.1.3 Necessary equilibrium estimates

We now include some equilibrium estimates from cluster expansions at suf-

ficiently large q, which are adaptations of the necessary low-temperature Ising

equilibrium estimates of [30] to the setting of the critical FK model with large q.

For any φ ∈ (−π/2, π/2), the strip Sn = J0, nK×J−∞,∞K has (1, 0, φ) boundary

conditions denoting wired on ∂Sn ∩ J0, nK×y ≤ x tanφ and free elsewhere. Then

I is the set of all order-disorder interfaces (bottom-most dual crossings from

(0, 0) ←→ (0, n tanφ)). Recall the definition of the cigar-shaped region Uκ,d,φ

from (7.1.2) and call Irκ,d,φ ⊂ I the subset of interfaces that are not contained in

the cigar-shaped region.

Proposition 7.1.6. Consider the critical FK model on Sn and fix a δ > 0. There

exists some q0 such that for all φ ∈ [−π2

+ δ, π2− δ], there exists c(d, φ) > 0 such

that for every q ≥ q0 and every κ > 0,

π1,0,φSn (Irκ,d,φ) . n2 exp(−cn2κ) .

Proof. We use an extension of [30, §4] to the framework of the FK/Potts models

in the phase coexistence regime by [72]. The specific case of [72, §5] states the

following: consider the critical FK model on the strip Sn = J0, nK × J−∞,∞K

with (1, 0, φ) boundary conditions. Then for every q ≥ q0 and every d, κ > 0, and

232

φ ∈ (−π2, π

2),

limn→∞

1

nlog

∑I∈Ird,κ,φ

π1,0,φSn (I)∑

I∈I π1,0,φSn (I)

= 0 .

A straightforward adaptation of the proof of [72, Proposition 5] to the form of [30,

Proposition 4.15] in fact yields the very large deviation bound we desire. A specific

case is when φ = 0: there exist some q0, c > 0 such that for all q ≥ q0 and every

a ≥ 0,

π1,0,φ=0Sn (|H| ≥ a) . n2 exp(−ca2/n) , (7.1.5)

where |H| denotes the maximum vertical distance of an edge e in the interface to

the x-axis. Though the result of [72] is written with the Potts model in mind and

thus with integer q, the cluster expansion and all the results hold with noninteger

q as well.

We now prove the following estimate on FK interfaces near a repulsive boundary.

Proposition 7.1.7. Fix c > 0 and consider the critical FK model on R = J0, nK×

J0, `K for c√n log n ≤ ` ≤ n with boundary conditions (1, 0) denoting 1 on ∂n,e,wR

and 0 on ∂sR. Let H be the maximum vertical height of the order-disorder interface.

There exist constants q0, c′ > 0 such that, for all q ≥ q0 and every 0 ≤ a ≤ `,

π1,0R (H ≥ a) . n2 exp

(−c′ a2/n

).

Proof. Fix the a ≤ ` from the statement of Proposition 7.1.7. Denote by (1, 0, ∗)

boundary conditions that are still wired (resp. free) on the intersection of ∂Sn and the

upper (resp. lower) half plane, but now also free on all of Sn ∩ (x, y) : y ≤ −a/2

233

(this induces a free bottom boundary on the semi-infinite strip starting from

y = −a/2).

Clearly π1,0Sn π1,0,∗

Sn . Then by Eq. (7.1.5), there exists A(q) > 0 such that with

probability bigger than 1 − An2 exp(−ca2/4n), the interface under π1,0Sn does not

touch the line y = −a/2, and therefore, with that probability, there is a horizontal

dual-crossing of Sn contained entirely above y = −a/2. Via the grand coupling,

since this is a decreasing event, the same horizontal dual-crossing would be present

under π1,0,∗Sn and we expose the bottom most horizontal dual-crossing above the line

y = −a/2 and couple the configurations above it.

At the same time, using (7.1.5), we have that under π1,0Sn , with the same

probability, the maximum y-coordinate of the interface does not exceed a/2. If we

have coupled the two configurations above a bottommost dual-crossing above the

line y = −a/2, the same would be true of the interface under π1,0,∗Sn . Thus, taking a

union bound,

π1,0,∗Sn (H ≥ a) ≤ 2An2 exp(−ca2/4n) .

Using the monotonicity of the FK model, and denoting by R′ the vertical

translate of R by −a/2, we obtain π1,0R′ π1,0,∗

Sn (ωR′); together with the fact that

H ≥ a is a decreasing event, for c′ = c/4,

π1,0R (H ≥ a) ≤ 2An2 exp(−c′a2/n) ,

where now, the interface is again between (0, 0) and (0, n).

Now for any fixed ε > 0, consider the rectangle V` = J0, nK×J0, `K with ` ≥ 4n12

+ε,

and (1, 0,∆) denoting free boundary conditions on ∂V ∩ (x, y) : y ≥ 2n12

+ε and

∆ = 0 × Jn2− n3ε, n

2+ n3εK and wired elsewhere. Denote the four points at which

234

the boundary conditions change by (w1, w2) ∈ ∂wV × ∂eV , z1, z2 ∈ ∂sV with z1 to

the left of z2. Let C1 and C2 be the blocks J0, n2K×J0, `K and Jn

2K×J0, `K respectively.

The main equilibrium estimate we use in the sequel reads as follows.

Proposition 7.1.8. Let Γ = ω : wiC∗i←→ zi, i = 1, 2. There exists q0 > 0 so

that the following holds. For every q ≥ q0 there exists c = c(q) > 0 such that the

corresponding critical FK model on V` satisfies

π1,0,∆V`

(Γc) . e−cn3ε

.

Proof. This corresponds to Claim 3.10 proven in the appendix of [70] for ` = n12

+ε

in the setting of the low-temperature Ising model using the cluster expansion

of [30] and the analogues of Propositions 7.1.4 and 7.1.5 (Propositions 4.15 and

Theorem 4.16 of [30] respectively), but the extension to larger ` is immediate. We

sketch the proof of [70] before justifying its extension to the current setting.

(a) Via a surface tension estimate analogous to (7.1.4), with probability 1 −

exp(−cn3ε), the interfaces connect wi ←→ zi instead of w1 ←→ w2. In order

to then claim that the two interfaces are confined to the right and left halves

of V , it is shown in the appendix of [70] that with high probability the two

interfaces do not interact, via the exponential decay of the Ising cluster weights

in cluster lengths.

(b) The next step in [70] was to show that the interface does not deviate farther

than nε from the east of z1. The complication in the Ising setup was that the

plus boundary on ∂sV produced a repulsive force on the interface so neither

Proposition 4.15 nor Theorem 4.16 of [30] were directly applicable.

235

(c) To circumvent the problem that ∂sV is not sufficiently far from z1 to contain

the cigar-shaped region, the region V is extended in [70] to V ∪ J0, n/2−n3εK×

J−n, 0K with appropriate boundary conditions. Thereafter, the proof concludes

by lower bounding the weight of all interfaces between w1 and z1 that do not

interact with the extension of V , repeatedly using Theorem 4.16 of [30]: the

key to this lower bound consists of stitching cigar shaped regions of increasing

length, all sufficiently far from the extension of V and lying above the straight

line connecting w1 to z1.

(d) If the extension is accounted for, an estimate of the form of Theorem 4.16 of [30]

with the appropriate angle φ implies that with high probability, the interface

does not deviate far to the east of z1, thus stays bounded away from ∂eC1.

For more details on these arguments, see [70, Appendix A]. With Propositions

7.1.4–7.1.6, steps (a)–(d) carry through in the setting of the critical FK model with

large q, proving Proposition 7.1.8. Notice that the long-range interactions of the

FK model are irrelevant to this situation where all boundary conditions used are

completely free or completely wired and cannot be perturbed.

7.2 Upper bounds under free boundary condi-

tions

In this section, we use the sharp bounds on interface fluctuations, obtained in

Section 7.1, along with the dynamical scheme of [63, 70] to prove sub-exponential

mixing for the FK Glauber dynamics under free and wired boundary conditions

at large q and p = pc(q). This in turn implies sub-exponential mixing for the

236

Swendsen–Wang dynamics with free or monochromatic boundary conditions when

q is large enough and β = βc(q). Because of the long-range interactions of induced

FK boundary conditions, we need to again modify FK boundary conditions to

disconnect these bridges at each step of the recursion. For that, let us define the

following boundary modification, for distributions over boundary conditions.

Definition 7.2.1 (boundary modification). If P is a distribution over boundary

conditions on ∂Λ, ∆ ⊂ ∂Λ, we let P∆ be the distribution over boundary conditions

ξ′ obtained by first sampling a boundary condition ξ ∼ P then setting ξ′ to be

the segment modification ξ∆ as defined in Definition 4.2.1; this induces a coupling

(ξ, ξ′) ∼ (P,P∆). E.g., if ∆ = ∂Λ then ξ′ = 0, and if ∆ consists of a single vertex v

and ξ is induced by a configuration where every boundary vertex is connected to v

and to no other boundary vertex, then ξ′ would be wired on ∂Λ− v.

7.2.1 The recursive scheme

Throughout this subsection, let P be a distribution over FK boundary conditions

on Λn,m and E the corresponding expectation. For ξ ∼ P, we say that

APn,m(t, δ) holds if max

ω0∈0,1E[∥∥∥P t(ω0, ·)− πξΛn,m

∥∥∥tv

]≤ δ .

Using this notation, the following corollary is a consequence of Lemma 2.2.13.

Corollary 7.2.2. Consider Λn,m with boundary conditions ξ ∼ P and ∆ ⊂ ∂Λn,m

such that |V (∆)| n3ε, with P∆ defined as in Definition 7.2.1. If for some t, δ,

maxω0∈0,1

E∆[‖P t(ω0, ·)− πξΛn,m‖tv

]≤ δ ,

237

then APn,m(t′, δ′) holds with δ′ = 8δ + exp(−ecn3ε

) and t′ = exp(cn3ε)t for some

c(q) > 0 independent of ∆ and ξ. Similarly, APn,m(t, δ) implies,

maxω0∈0,1

E∆[‖P t′(ω0, ·)− πξΛn,m‖tv

]≤ δ′ .

Before proving the main theorem, we fix an ε > 0 and prove a recursive scheme

that yields a mixing time bound on rectangles with side lengths n× n 12

+ε for n of

the form n ∈ 2kk∈N. We remark that as in [70], this is a technical assumption

that is not requisite to the upper bound, (see Remark 3.12 of [70]).

For the base scale of the recursion, we use a the following: it is a consequence

of a trivial mixing time upper bound that is exponential in the longer side-length

of a rectangle, that holds independently of the boundary conditions (e.g., pay an

exponential in the perimeter cost to make the boundary conditions free, then use

the canonical paths estimate of Theorem 2.2.9).

Proposition 7.2.3. There exists c = c(q) > 0 such that for every n, for every q,

for the FK Glauber dynamics, APn,m(t, exp[−te−c(n∨m)]) holds independent of P.

An intermediate step to proving Proposition 7.0.2 is proving analogous bounds

for rectangles with free-at-infinity boundary conditions on three sides and wired-at-

infinity on the fourth.

Definition 7.2.4. A distribution P over boundary conditions on Λn,m is in D(Λn,m)

if it is dominated by π0Z2 on ∂n,e,wΛn,m and dominates π1

Z2 on ∂sΛn,m.

We say that An,m(t, δ) holds if APn,m(t, δ) holds for every P ∈ D(Λn,m).

The main estimate for our recursion on increasing rectangles is the following.

238

Proposition 7.2.5. For the critical FK Glauber dynamics with q large enough

on Λn,m, the following holds: for any m ∈ Jn12

+ε, nK and α ∈ (1, 2), there exist

c1, c2 > 0 such that for every t, δ,

An,m(t, δ) =⇒ An,bαmc(

2ec2n3ε

t , c1(δ + e−c2n2ε

+ n2t−c2)), (7.2.1)

and for every m n12

+ε there exist c1, c2 > 0 such that for every t, δ,

An,m(t, δ) =⇒ A2n,m

(3ec2n

3ε

t , c1(δ + e−c2n3ε

)). (7.2.2)

Before proving the implications in Proposition 7.2.5, we first prove two easy but

important consequences.

Corollary 7.2.6. There exists q0 such that, for every q ≥ q0, there exist c, c′ > 0

such the following holds. If n ∈ 2kk∈N is sufficiently large, then the statement

An,n1/2+ε(exp[cn3ε], exp[−c′n2ε]) holds for critical FK Glauber dynamics on Λn,n1/2+ε.

Proof. Choose n0 nε and let t0 = exp(c′nε) for some constant c′ > 0 large

enough that An0,n

1/2+ε0

(t0, δ0) holds with δ0 = exp(−c′nε), noting that such a

choice of c′ exists by Proposition 7.2.3. Applying (7.2.1) followed by (7.2.2) with

m = n12

+ε and α = 212

+ε allows one to, for any n, express the mixing time of a

2n × (2n)12

+ε box in terms of that of a n × n 12

+ε box. Starting with the scale n0

and repeating this step log2 n times implies that there exists c > 0 fixed such that

An,n1/2+ε(exp(cn3ε), c exp[−n−2ε/c]) holds.

We now use the bound on n× n 12

+ε rectangles to obtain mixing time bounds on

the n×m rectangle with boundary conditions that are disordered on three sides

and ordered on the fourth.

239

Corollary 7.2.7. Consider the critical FK Glauber dynamics on Λn,m for m ∈

Jn12

+ε, nK and boundary conditions ξ ∼ P. Then there exists q0 > 0 such that for

all q ≥ q0 there exists a constant c = c(m, q) > 0 such that for large enough n, for

every P ∈ D(Λn,m),

maxω0∈0,1

E[‖P t?(ω0, ·)− πξΛn,m‖tv

]. e−cn

2ε

,

for t? = exp(cn3ε), and in particular if tmmix is the corresponding mixing time,

P(tmmix ≥ t?) . e−cn2ε

.

Proof. Choose an α ∈ (1, 2) such that αkn12

+ε = m for some integer k logm.

Then let hj = bαjn 12

+εc and let Λj = Λjn = J0, nK× J0, hjK so that Λk = Λn,m.

We prove the above by induction on j ∈ J0, kK for n× hj rectangles, showing

that

maxω0∈0,1

E[‖P thj (ω0, ·)− πξΛn,hj ‖tv

]≤ (cj1 + 2jc1)e−c2n

2ε

, (7.2.3)

where c1, c2 are the constants of (7.2.1) for m = hj, and

thj = 2jhjc2(1+n3ε) .

The base case j = 0 is given by Corollary 7.2.6, and if (7.2.3) holds for some

fixed j ∈ J0, k − 1K, then an application of (7.2.1) immediately implies it for j + 1.

The observations that j ≤ log n and hj ≤ n allow us to choose slightly different

constants to obtain the first inequality of Corollary 7.2.7. The triangle inequality

and Eq. (2.2.7) can then be used to boost the bound on d1(t) ∨ d0(t) to a bound

on d(t), so that Markov’s inequality implies the second inequality.

240

B

A

Q

n

bαmcm

1

1

1

t

1

ν1

1

1

η

t

νη2

η

Figure 7.1: Setup for the proof of Eq. (7.2.1) starting from wired initial conditions.

We now prove Proposition 7.2.5 from which the above corollaries follow. The

proof of Proposition 7.0.2 then follows from Corollary 7.2.7 using similar techniques

(see §7.2.2)

Proof of Eq. (7.2.1). Fix any P ∈ D(Λn,bαmc) and observe that the proof is

independent of this choice of P. Consider the quantity, E[‖P t(ω0, ·)− πξΛn,bαmc‖tv]

for ω0 = 0, 1.

(i) Wired initial conditions. Begin with the case when ω0 = 1. Let A,B be two

copies of Λn,m with A translated upwards by b(α− 1)mc such that Q := A ∪B =

Λn,bαmc and A ∩B is the middle rectangle in Qn of thickness m.

In order to compensate for the long-range interactions of the FK model, that are

not present in the setting of [70], we force a set of boundary edges to be free (in a

manner similar to part 2 of the proof of Theorem 3.2 of [70]) to “disconnect” B from

A. Consider the boundary condition ξ′, a modification of ξ ∼ P on ∆ = ∆s ∪∆n

241

for,

∆s =(x, y) ∈ ∂e,wA : y ≤ b(α− 1)mc+ n3ε ,

∆n =(x, y) ∈ ∂e,wB : y ≥ bm− n3εc ,

according to Definition 7.2.1. By Corollary 7.2.2 it suffices, up to new choice

of constants c1, c2 to show that the FK Glauber dynamics under P∆ on A ∪ B

satisfies (7.2.1).

Denote by P the transition kernel of the censored dynamics (Xs)s≥0 started

from the all wired configuration, only accepting updates in block A up to time t,

resetting all edge values in B to 1 at time t then only accepting updates in block

B from time t to time 2t (observe that as in Lemma 3.4 of [70], by Theorem 2.2.7,

resetting all edge values to 1 only slows mixing). Let ν1 denote the distribution

after time t on A and let νη2 denote the distribution after time 2t of configurations

on B given that at time t the configuration on B was set to 1 and the boundary

condition on Bc was η (see Fig. 7.1).

The monotonicity of the FK model along with Theorem 2.2.7 yields,

dξ′

1 (2t) ≤ ‖P 2t(1, ·)− πξ′

Q‖tv . (7.2.4)

Now we aim to show that E∆[‖P 2t(1, ·)−πξ

′

Q‖tv]≤ δ′ where we let δ′ be the second

argument in the right hand side of (7.2.1). For R = A,B we denote by πξ′,ηR the

modified stationary distribution with η boundary conditions on Q−R.

To simplify the notation, throughout the rest of this section, we let ‖µ− ν‖R

denote ‖µR − νR‖tv. Also, for any R, ξ and any random variable X, let πξR(X)

denote the expectation of X under πξR. By the Markov property and the triangle

242

inequality,

E∆[‖P 2t(1, ·)− πξ

′

Q‖tv]≤ E∆

[‖ν1 − πξ

′,1A ‖Bc

]+ E∆

[‖πξ

′,1A − πξ

′

Q‖Bc]

+E∆[‖πξ

′

Q − πξ′,0Q ‖Bc

]+ E∆

[πξ′,0Q (‖νη2 − π

ξ′,ηB ‖tv)

].

(7.2.5)

We begin by bounding the first and fourth terms which are easier, then use

the equilibrium estimates of the cluster expansion to bound the third term in

Lemma 7.2.8, analogous to which the second term can be bounded. First observe

that with probability 1− exp(−cn3ε) for some c > 0, the boundary conditions on

A are sampled from a distribution in D(A). The concern is that the wired initial

configuration may add connections to ∂n,e,wA via the long-range FK interactions.

Such an effect on the boundary conditions on A is impossible if there are no

boundary connections from ∂e,wAc to ∂n,e,wA. Because of the modification on

∆s such a connection would require a connection of length at least n3ε under P

and therefore also in the free phase, which has probability less than exp(−cn3ε)

(see Eq. (2.1.2)). If no such connection exists along the boundary, the boundary

conditions on ∂n,e,wA are sampled from a measure dominated by π0Z2 because the

modification of Definition 7.2.1 only removes connections. From now on, paying a

cost of exp(−cn3ε), we assume the decreasing event that this is the case.

Then by the assumption that An,m(t, δ) holds, the first term in (7.2.5) is smaller

than δ. The observation that P∆ P on ∂e,wQ, implies that for any decreasing f

only depending on ∂n,e,wB,

E∆[πξ′,0Q (f)

]≥ π0

Z2(πξQ(f)) = π0Z2(f) , (7.2.6)

243

so that the πξ′,0Q -averaged distribution on boundary conditions on B is in D(B),

the statement An,m(t, δ) applies, and the fourth term in (7.2.5) is also bounded

above by δ.

We now turn to the second and third terms of (7.2.5), which can be bounded

similarly, and thus we only go through the details of the third term:

Lemma 7.2.8. There exists c(q) > 0 such that

E∆[‖πξ

′

Q − πξ′,0Q ‖Bc

]. e−cn

2ε

.

Proof. We bound the total variation distance by proving that under the grand

coupling of the two distributions on Bc, they agree with probability 1 = e−cn2ε

.

Let ∂±(Q) denote the two connected components of ∂Q−∆n above and below ∆n

respectively. We break up the expectation into an average over Γ1, the set of ξ′ in

which there does not exist a pair (x, y) ∈ ∂+Q× ∂−Q such that xξ′←→ y (i.e. they

are in the same boundary component), and Γc1. By Eq. (2.1.2) of Theorem 2.1.2

and a union bound over pairs of boundary vertices, there exists a constant c′(q) > 0

such that

P∆(Γc1) ≤ 16n2e−cn3ε

. exp(−c′n3ε) , (7.2.7)

For all such ξ′, we use the worst bound of 1 on the total variation distance.

Suppose now that Γ1 holds and observe that this is a decreasing event so P∆(· |

Γ1) P∆. Let Γ2 denote the decreasing event that the interface (bottom-most

horizontal dual crossing) of Q is contained entirely below ∂s(Bc). Let Γ3 be the

decreasing event that there does not exist any vertex x ∈ ∂−e Q such that x←→ ∂nB,

244

n n

n

Q

En(Q)

E′n(Q)

Figure 7.2: Bounding Γ2: if a dual-open circuit (purple) exists under π0Z2 in

E′n(Q)−Q then the boundary conditions on ∂n,e,wQ are dominated by those underfree on E ′n(Q). The wired boundary conditions on ∂s(En(Q)) then also dominate thewired-at-infinity boundary conditions on ∂sQ allowing us to dominate the interface(blue) in Q by that in En(Q).

and there does not exist any y ∈ ∂−wQ such that y ←→ ∂nB.

By monotonicity and the domain Markov property, for ξ′ ∈ Γ1, if Γ2 ∩ Γ3 holds,

it is the case that the boundary conditions on ∂n,e,wBc will not have been affected

by the updates on A ∩B as the interface and all its long-range interactions with

∂Q would be confined to A ∩B. Then one could reveal all boundary components

of ∂e,s,wB so that they are all confined to B and by monotonicity under the grand

coupling the two distributions would be coupled on Bc. As a result,

E∆[‖πξ

′

Q − πξ′,0Q ‖Bc

∣∣ Γ1

]≤ E∆

[πξ′,1Q (Γc2 ∪ Γc3)

∣∣ Γ1

]. (7.2.8)

We bound the two probabilities separately and take a union bound. To bound the

245

probability of Γc2, consider the enlarged rectangle,

En(Q) = J−n, 2nK× J0, n+ αmK ⊃ Q , (7.2.9)

with (0, 1) boundary conditions denoting wired on ∂sEn(Q) and free elsewhere.

By Definition 7.0.1 we sample ∂n,e,wQ separately and then ∂sQ. First observe

that by Eq. (2.1.2), with π0Z2-probability 1− e−cn, there is a dual circuit between

Q and its enlargement by n in all four sides E′n(Q). In that case, the boundary

conditions on Q are dominated by those with free on ∂E′n(Q) (see Figure 7.2). We

can subsequently dominate the boundary conditions on ∂sQ by making them all

wired and extending them all the way across En(Q) to obtain that there exists

c > 0 such that

E∆[πξ′,1Q (Γc2)

∣∣ Γ1

]≤ π0,1

En(Q)(Γc2) + e−cn .

By Proposition 7.1.7, we deduce that π0,1En(Q)(Γ

c2) . exp(−cn2ε) for some c > 0.

We now bound the probability of Γc3.

Claim 7.2.9. There exists c = c(q) > 0 such that for every ξ′ ∈ Γ1,

πξ′,1Q (Γc3 | Γ2) . e−cn

3ε

+ e−cn12 +ε

+ e−cn .

Proof. Under Γ2, by monotonicity, we can only worsen our bound on the probability

of Γc3 by replacing the boundary conditions on Q by wired on ∂−Q− ∂sQ, free on

∂sQ, and ξ′ elsewhere.

Let Q = J0, nK× J0, 2mK ⊃ Q with boundary conditions free on ∂s,nQ∪∆n, and

wired elsewhere. By the exponential decay of correlations in the free phase, with

πQ-probability 1− e−cm, the measure this induces on Q dominates the boundary

246

“1”

“0”

0“0”

∆n

B

Figure 7.3: The event Γ2 is the event that the blue cluster does not climb toohigh. Under Γ2, there is a horizontal dual-crossing of B immediately adjacent theblue interface, but the boundary of the red component may have been perturbedby connections in the blue shaded region. The event Γ3 is the event that the redcomponent then does not then climb above ∆n. Boundary conditions “1” (“0”)indicate those that dominate (are dominated by) wired-at infinity (free-at-infinity).

conditions under Γ2 on Q.

Controlling the probability of Γc3 can now be expressed in a manner similar to

the equilibrium bound, Proposition 7.1.8. As is standard in such problems, (see,

e.g., the appendix of [70]), we can up to an error of e−cn separate the left and right

interfaces (see Proposition 7.1.5 whence the probability that they interact is a large

deviation of order n), and just consider Q with free boundary conditions now on

all of ∂eQ also.

Then extend the northern boundary of Q to make Q symmetric about ∆n and

call the new domain Q′. We can, using monotonicity, let its boundary conditions

(1, 0,∆n) be free on ∂Q′ ∩ (x ≥ m12

+ε ∪∆n) and wired elsewhere.

At this point, we apply Proposition 7.1.8 with ` = n (up to a π2-rotation and

247

a rescaling of m to n/2) to obtain the desired bound: if in the new domain, the

boundary points are denoted by w1, w2 ∈ ∂nQ′ × ∂sQ

′ and z1, z2 ∈ ∂eQ′, and

Cii∈n,s are the north or south halves of Q′ respectively, Proposition 7.1.8 implies

that there exists c > 0 such that for large enough n,

π1,0,∆n

Q′

( ⋂i=n,s

wiCi←→ zi

)≤ e−cn

3ε

.

Putting everything together, we conclude that if Γ′ is the event that under

π1,0,∆n

Q′ , the two interfaces are contained in bottom and top halves of Q′ respectively,

then there exists c = c(q) > 0 such that, for large enough n,

π1,0,∆n

Q(Γ′c) ≤ 2e−cn

3ε

+ 2e−cm + 2e−cn ,

Monotonicity and n12

+ε ≤ m ≤ n imply the bound on E∆[πξ′,1Q (Γc3 | Γ2) | Γ1].

By union bounding over the errors that arise from each of the Γi not occurring,

and otherwise conditioning on their occurrence, we obtain

E∆[‖πξ

′

Q − πξ′,0Q ‖Bc

]. e−cn

2ε

,

as desired.

The corresponding bound on the second term of (7.2.4) is the same up to

changes of scale corresponding to working with distributions on configurations on

A, not Q. Because of the modification on ∆s, as remarked earlier, with probability

1 − exp(−cn3ε), the boundary conditions on A are in D(A). This event is a

decreasing event so it only increases the probability of Γi for i = 1, 2, 3. Because

248

α > 1, the middle rectangle is at least order n12

+ε so the bound on the interface

touching ∂s(Bc) also still holds. Combined with (7.2.4) and the bounds on the

other terms in (7.2.5), we conclude that for some c1, c2 > 0 and all large enough n,

E∆[dξ′

1 (2t)]≤ 2δ + 2c1e

−c2n2ε

.

(ii) Free initial configuration. Consider the dynamics started from ω0 = 0. Let

P denote the transition kernel of the censored dynamics that only updates edges

in B until time t at which point all edges in A are reset to 0 and the dynamics

subsequently only updates edges in A until time 2t. Let νη2 denote the distribution

obtained between times t and 2t given boundary conditions η on Ac and initial

configuration 0 on A. Let

∆ = (x, y) ∈ ∂e,w(Ac) : y ≥ b(α− 1)m− n3εc ,

so that again by Theorem 2.2.7 and Corollary 7.2.2 it suffices to prove the desired

implication under E∆ for the P dynamics.

The Markov property and the triangle inequality together imply,

E∆[‖P 2t(0, ·)− πξ

′

Q‖tv]≤ E∆

[‖P t(0, ·)− πξ

′,0B ‖Ac

]+ E∆

[‖πξ

′,0B − πξ

′

Q‖Ac]

+E∆[πξ′

Q(‖νη2 − πξ′,ηA ‖tv)

]. (7.2.10)

We can bound the first term by δ by assumption and the observation that the free

initial configuration does not change the boundary conditions on B. The second

term can be bounded using the same approach as the proof of Lemma 7.2.8, where

now Ac is shorter than the rectangle in Lemma 7.2.8 but still & n12

+ε. Via an

249

application of Lemma 7.2.8, up to an error of exp(−cn2ε) for c > 0 fixed, the open

component of ∂sB and its perturbations to the boundary of A ∪ B do not reach

∂nAc so the grand coupling couples the two distributions on Ac.

In that case, the measure on ∂A is dominated by π0Z2 and therefore by (2.1.2),

up to an error of e−cm the entirety of Bc is disconnected from Ac. By monotonicity

and the fact that m ≥ n12

+ε, we obtain that there exists c > 0 such that

E∆[‖πξ

′,0B − πξ

′

Q‖Ac]. e−cn

12 +ε

+ e−cn2ε

.

It remains to bound the third term in (7.2.10) following the approach of [70].

Lemma 7.2.10. There exist constants c, c′ > 0 such that

πξ′(‖νη2 − π

ξ′,ηA ‖tv) . e−cn

2ε

+ e−c′` .

Proof. Using the bound on total variation by the probability of disagreement under

a maximal coupling, together with monotonicity and a union bound, write

πξ′(‖νη2 − π

ξ′,ηA ‖tv) ≤

∑e∈E(A)

πξ′

Q

(νη2 (e /∈ ω)− πξ

′

Q(e /∈ ω)).

For any e ∈ E(A), consider K` = e+ J−`, `K2∩A. Denote by νη2,` the distribution

obtained by the dynamics in K` with boundary conditions given by (ξ′, η) on

∂K` ∩ ∂A and 0 elsewhere. Then via a very rough mixing time estimate akin to

Proposition 7.2.3, there exists c > 0 such that

νη2 (e /∈ ω)− πξ′

Q(e /∈ ω) ≤ e−te−c`

+(πξ′,ηK`

(e /∈ ω)− πξ′

Q(e /∈ ω)).

250

Absorbing a 2n2 for the maximum number of edges in A, it suffices to prove there

exist constants c, c′ > 0 such that for every e ∈ E(A),

E∆[πξ′

Q(πξ′,ηK`

(e /∈ ω)− πξ′,ηA (e /∈ ω))

]. e−c

′` + e−cn2ε

.

For any fixed e ∈ E(A) let Γc := e K`←→ ∂K` ∩ Ao. By the FKG inequality,

πξ′,ηA (e /∈ ω | Γ) ≥ πξ

′,ηK`

(e /∈ ω) ,

so it suffices to check that

E∆[πξ′

Q(πξ′,ηA (Γc))

]= E∆

[πξ′

Q(Γc)]. e−c` + e−cn

2ε

.

Proving this is very similar to proving the bound on the probability of Γc2 in

the proof of Lemma 7.2.8 as shown in Figure 7.2. For some c > 0, up to an error of

e−cn, we replace E∆[πξ′

Q(Γc)] with π0,1En(Q)(Γ

c) where En(Q) is the enlarged rectangle

defined in (7.2.9). Then by Proposition 7.1.7, for some other c > 0 with probability

1−exp(−cn2ε), the bottom-most horizontal crossing stays below ∂sA at which point

it suffices to consider π0En(Q)(Γ

c) because in that case, ∂sEn(Q) would be completely

disconnected from K`. But Eq. (2.1.2) and monotonicity imply that there exists

c > 0 such that

π0En(Q)(Γ

c) . e−c` ,

at which point a union bound over the two errors concludes the proof.

Choosing ` = dc−1 log te in Lemma 7.2.10 and union bounding over all e ∈ E(A)

251

yields that there exists a new c > 0 such that for sufficiently large n,

E∆[πξ′(‖νη2 − π

ξ′,ηA ‖tv)

]. t−c .

Combined with the bounds on the first and second terms of (7.2.10) and Theo-

rem 2.2.7 we see that there exist c1, c2 > 0 such that for large enough n,

E∆[‖P 2t(0, ·)− πξ

′

Q‖tv]≤ 2δ + c1e

−c2n2ε

+ 2n2t−c2 ,

which combined with part (i) of the proof, allows us to conclude the proof of (7.2.1).

We now prove the second implication to complete the proof of Proposition 7.2.5.

Proof of Eq. (7.2.2). Fix any P ∈ D(Λ2n,m) and observe that the proof is inde-

pendent of this choice of P. Consider the quantity, E[‖P t(ω0, ·) − πξΛ2n,m‖tv] for

ω0 = 0, 1.

(i) Wired initial configuration. Divide Q := Λ2n,m into,

A =Λn,m + (bn/2c, 0) , B = Λn,m ∪ Λn,m + (n, 0) , C = n × J0,mK .

Let Be, Bw be the two connected components of B so that the dynamics on B does

not update any of the edges of C ⊂ ∂B. Let ∆ = ∆s ∪∆n ∪∆e ∪∆w where

∆s = (x, y) ∈ ∂sA : |x− n| ≤ n3ε , ∆n = (x, y) ∈ ∂nA : |x− n| ≤ n3ε ,

∆e = (x, y) ∈ ∂nA : x ≤ n/2 + n3ε , ∆w = (x, y) ∈ ∂nA : x ≥ 3n/2− n3ε .

We first observe that, as before, by Corollary 7.2.2 and the size of |V (∆)|, it

252

suffices up to new choice of constants in (7.2.2), to prove the implication under P∆

as given by Definition 7.2.1. By monotonicity, Theorem 2.2.7, and Corollary 7.2.2,

up to another change of constants c2, c3, it suffices to prove,

maxω0∈0,1

E∆[‖P 2t′(ω0, ·)− πξ

′

Q‖tv]≤ cδ′

with t′ = exp(cn3ε)t and δ′ = 8δ+ exp(−ec′n3ε) for c, c′ > 0 given by Corollary 7.2.2

and P a censored dynamics. We begin with the situation in which ω0 = 1 and let

P be the transition kernel of the following censored dynamics: for the first time

interval [0, t′), only accept updates from A then at time t′ change all edges interior

to B to 1 and only updates edges interior to B until time 2t′. As before, let ν1

denote the distribution after time t′ on A and let νη2 denote the distribution after

time 2t′ of configurations on B given that at time t′ all edges in B are reset to 1

and the configuration on C was η.

The triangle inequality and Markov property together imply that,

E∆[‖P 2t′(1, ·)− πξ

′

Q‖tv]≤ E∆

[‖ν1 − πξ

′,1A ‖C

]+ E∆

[‖πξ

′,1A − πξ

′

Q‖C]

+E∆[‖πξ

′

Q − πξ′,0Q ‖C

]+ E∆

[πξ′,0Q (‖νη2 − π

ξ′,ηB ‖tv)

].

(7.2.11)

Here, boundary conditions of the form (ξ′, η) denote a boundary condition that

agrees with ξ′ on the intersection of the boundary of the domain and ∂Q, and takes

on boundary conditions η elsewhere. As in part (i) of the proof of (7.2.1), we observe

that because of the modification on ∆e ∪∆w with P-probability 1− exp(−cn3ε),

the boundary conditions on ∂nA are dominated by π0Z2 in spite of the wired

initial configurations on Q − A. In what follows, we assume—paying an error

253

of exp(−cn3ε)—that this decreasing event holds. Observe also that the wired

initial configuration can only make ∂sA more wired, and thus those boundary

conditions will continue to dominate the marginal of π1Z2 . Along with self-duality

and Corollary 7.2.2, this implies that the first term in (7.2.11) is bounded above by

δ′.

The fourth term can be bounded as follows: observe that the distribution

over boundary conditions (ξ′, η) on B under P∆(ξ′)πξ′,0Q (η) coincides with the ∆s

modification of P∆n,e,w(ξ′)πξ′,0Q (η). Because the boundary conditions on ∂nA are

dominated by π0Z2 , an argument like (7.2.6) implies P∆n,e,w(ξ′)πξ

′,0Q (η) π0

Z2(η) on

C. Thus, by Corollary 7.2.2, the third term is bounded above by 2δ′, where the

factor of 2 comes from the fact that B consists of two independent copies of Rn.

We used the fact that the configuration on E(C) is dominated by the partition of

C induced by the FK configuration under πξ′,0Q .

It remains to bound the second and third terms of (7.2.11), which can be treated

similarly so we only go through the bound of the former.

Lemma 7.2.11. There exists c = c(q) > 0 such that

E∆[‖πξ

′,1A − πξ

′

Q‖C]. e−cn

3ε

.

Proof. We bound the total variation distance on C by bounding the probability

that samples from the two distributions agree under the grand coupling.

Following the proof of Lemma 7.2.8, we first define the event Γ1 as the set of

ξ′ in which there exists no pair (x, y) ∈ (∂nBe − ∆n) × (∂nBw − ∆n) such that

xξ′←→ y where

ξ′←→ denotes that x, y are in the same boundary component. We

split up E∆[‖πξ′,1A − πξ

′

Q‖C ] into an average over those ξ′ ∈ Γ1 and those in Γc1.

254

“1” “1”

“1” “1”

∆s

∆n

CA A

“0” “0”

Figure 7.4: The modification analogous to Figure 7.3 for the second step of therecursion (Eq. (7.2.2)).

As in Lemma 7.2.8, we obtain the bound E∆[Γc1] ≤ exp(−cn3ε) for some c > 0

using the fact that the boundary conditions on ∂nA are obtained from a measure

that is dominated by π0Z2 and Eq. (2.1.2) implies an exponential decay of connections.

For all such ξ′, we bound the distance between the two measures by 1.

Now consider, for any e ∈ E(C), E∆[‖πξ′,1A − πξ

′

Q‖tv | Γ1]. Define in analogy

with the proof of Lemma 7.2.8, the decreasing events Γ2 and Γ3:

Γ2 :=⋂

i∈e,w

C∗v(A ∩Bi) ,

and Γ3 is the event that there does not exist any x ∈ ∂nA ∩ Be −∆n such that

x ←→ ∂wBe and likewise, there does not exist any x ∈ ∂nA ∩ Bw − ∆n such

that x ←→ ∂eBw. Under Γ2 ∩ Γ3, we could expose all the wired components of

∂A−∆n −∆s to reveal an outermost dual circuit around C. By monotonicity and

domain Markov, the two distributions would be coupled under the grand coupling

past that dual circuit, so that for all ξ′ ∈ Γ1,

‖πξ′,1A − πξ

′

Q‖C ≤ πξ′,1A (Γ2 ∩ Γ3) .

255

Let A = J0, nK× J2mK, viewed as two copies of A stacked above one another.

Let (0, 1,∆s) boundary conditions on A denote those that are free on ∂nA and

∆s and wired on the rest of ∂A. By monotonicity and the exponential decay of

correlations in the free phase given by (2.1.2), we see that there exists c > 0 such

that

E∆[πξ′,1A (Γc2)

∣∣ Γ1

]≤ E∆s

[πξ′,1A (Γc2)

∣∣ Γ1

]≤ e−cm + π0,1,∆s

A(Γc2) .

Consider π0,1,∆s

A(Γc2) and let V = J0, nK× J0, 4mK, as in Proposition 7.1.8 with

` = 4m. Let (0, 1,∆s) boundary conditions on V denote those that are free above

y = 2m and on ∆s and wired elsewhere. Then it is clear by monotonicity and the

fact that Γc2 is an increasing event, that

π0,1,∆s

A(Γc2) ≤ π0,1,∆s

V (Γc2) .

Because m n12

+ε, we can apply Proposition 7.1.8 to see that there exists a new

c(q) > 0 such that π0,1,∆s

A(Γc2) . exp(−cn3ε).

We now turn to bounding πξ′,1A (Γc3 | Γ2) using the same approach as in the proof

of Claim 7.2.9. Under Γ2, there is a pair of vertical dual crossings in A which allow

us to replace, by monotonicity, the boundary conditions (ξ′, 1) by ones that we

denote (0, 1,∆n) which are free on ∂e,w,sA and also free on ∆n and wired elsewhere.

To make the comparison to the setting of Proposition 7.1.8, perturb the boundary

conditions more by extending the wired segments down along ∂e,wA a length n1/2+ε.

Up to a π-rotation, Proposition 7.1.8 with the choice of ` = m implies that the

two interfaces are confined to the left and right halves of A with probability

1− exp(−cn3ε) for some new c > 0, and sufficiently large n. Monotonicity implies

256

that for any ξ′ ∈ Γ1,

πξ′,1A (Γc3 | Γ2) . e−cn

3ε

,

and together with a union bound, there exists c′ > 0 such that for large enough n,

E∆[‖πξ

′,1A − πξ

′

Q‖C]≤ e−c

′n12 +ε

+ 2e−cn3ε

.

Combined with the prior bounds on the first and third terms in the right-hand

side of (7.2.11) and Theorem 2.2.7, for some c > 0,

E∆[‖P 2t′(1, ·)− πξ

′

Q‖tv]. δ′ + e−cn

3ε

.

(ii) Free initial configuration. The bound for the free initial configuration,

E∆[‖P 2t′(0, ·)− πξ

′

Q‖tv]. δ′ + e−cn

3ε

,

is nearly identical to the above bound with the following change: the censored

dynamics P only allows updates in block A until time t′ then resets all edge values

in B to 0 then only allows updates in B between time t′ and 2t′. In this case, we

can again write, using the same notation as before,

E∆[‖P 2t′(0, ·)− πξ

′

Q‖tv]≤ E∆

[‖ν1 − πξ

′,0A ‖C

]+ E∆

[‖πξ

′

Q − πξ′,0Q ‖C

]+E∆

[‖πξ

′,0A − πξ

′

Q‖C]

+ E∆[πξ′,0Q (‖νη2 − π

ξ′,ηB ‖tv)

].

(7.2.12)

The free initial configuration precludes the long-range interactions of the FK model

modifying the boundary conditions on A and thus they are still in D(A). The

257

bound on the first term then follows from the assumption, without appealing to

self-duality, and the other three bounds hold as for ω0 = 1. Combining the above

with part (i) holds, we see that (7.2.2), which with (7.2.1), concludes the proof of

Proposition 7.2.5.

7.2.2 Proof of Proposition 7.0.2

It now remains to extend Corollary 7.2.7 to boundary conditions that are

dominated by the free phase or dominate the wired phase on all four sides, to obtain

the desired bounds for free and monochromatic boundary conditions. Suppose

without loss of generality that P π0Z2 on Λ; the case P π1

Z2 follows from

self-duality. First consider the dynamics started from ω0 = 1. We wish to prove

that there exists c > 0 such that for t? = exp(cn3ε),

E[‖P t?(1, ·)− πξ

′

Λ‖tv]. e−cn

2ε

.

By an application of Corollary 7.2.2, up to errors that can be absorbed by adjusting

the constant c appropriately, we can modify, as in Definition 7.2.1, the boundary

conditions on ∆ = ∆n ∪∆s, where ∆n = ∆1n ∪∆2

n,

∆1n =0, n × J3n

4− n3ε, 3n

4K ,

∆2n =0, n × Jn

2, n

2+ n3εK ,

and ∆s is the reflection of ∆n across the line y = n/2, and consider dynamics on

Λ with the new measure P∆ on boundary conditions. Let Λ± denote the top and

258

bottom halves of Λ, respectively. Then,

E∆[‖P t?(1, ·)− πξ

′

Λ‖tv]≤ E∆

[‖P t?(1, ·)− πξ

′

Λ‖Λ−

]+ E∆

[‖P t?(1, ·)− πξ

′

Λ‖Λ+

].

We deal only with the first term since the second can be bounded analogously.

Let P be the dynamics that censors all updates not in Λn,3n/4. Then by Markov

property and triangle inequality, the first term can be bounded above by

E∆[‖P t?(1, ·)− πξ′,1‖Λ−

]+ E∆

[‖πξ

′

Λ − πξ′,1‖Λ−

],

where πξ′,1 denotes the stationary distribution on Λn,3n/4 with boundary conditions

that are wired on ∂nΛn,3n/4 and ξ′ elsewhere. First observe that because of ∆1n, with

probability 1− exp(−cn3ε), the wired initial configuration on Λ− Λn,3n/4 does not

affect the boundary conditions on Λn,3n/4 and therefore the boundary conditions

on it are, up to a π-rotation, in D(Λn,3n/4). At this cost, we assume this decreasing

event holds.

The second term can then be bounded as in Lemma 7.2.8 by exp(−cn2ε) for

some c > 0. We use ∆2n to disconnect the wired boundary condition on ∂nΛn,3n/4

from Λ−; the new choice of ∆ and modifications in the sizes of the boxes do not

affect the proofs.

The first term can be bounded via Corollary 7.2.7 for m = 3n/4 since, with

probability 1−exp(−cn3ε), the boundary condition on ∂e,s,wΛn,3n/4 is dominated by

the marginal of π0Z2 while on ∂nΛn,3n/4, it is all wired. Combining the two bounds

and doing the same for Λ+, implies there is a c > 0 such that for t? = exp(cn3ε),

E∆[‖P t?(1, ·)− πξ

′

Λ‖tv]. e−cn

2ε

.

259

For the dynamics started from the free initial configuration, for every e ∈ E(Λ),

` ∈ N, define K` = Λ ∩ e+ J−`, `K2. Let PK` denote the transition kernel of the

dynamics restricted to K` with (ξ, 0) boundary conditions denoting ξ on ∂K` ∩ ∂Λ

and free elsewhere. We claim that, for some c > 0,

E[‖P t?(0, ·)− πξΛ‖tv

]≤∑

e∈E(Λ)

E[P t?(1, e /∈ ω)− πξΛ(e /∈ ω)

]≤∑

e∈E(Λ)

E[‖P t?

K`(0, ·)− πξ,0K`‖tv

]+ e−c` + e−cn ,

for some c > 0. The first inequality is an immediate consequence of monotonicity.

The second follows from an argument similar to that in the proof of Lemma 7.2.10

where now we take a new enlargement, E′n(Λ) that enlarges Λ by n in the southern

direction also. We can replace, by Eq. (2.1.2), πξΛ by π0E′n(Λ) up to an error of e−cn as

before. Again using (2.1.2) in the free phase, up to an error of e−c`, we can replace

π0E′n(Λ) with πξ,0K` , noting that the distributions at e match if e is disconnected from

K` by a dual circuit.

We can then bound the sum in the right-hand side by Proposition 7.2.3: uni-

formly in ξ,e, tmix for K` is bounded above by ec` for some c > 0. Choosing

` = dc−1 log t?e n3ε and union bounding over all the errors yields, for some c > 0,

E[‖P t?(0, ·)− πξΛ‖tv

]. e−cn

2ε

,

as desired. An application of Markov’s inequality, the triangle inequality, and (2.2.7)

to

maxω0∈0,1

E[‖P t?(ω0, ·)− πξΛ‖tv

]. e−cn

2ε

,

260

implies that there exists some c > 0 such that for t? = ecn3ε

and n sufficiently large,

P(tmix ≥ t?) . exp(−cn2ε) as required.

261

Chapter 8

Boundary conditions in

coexistence regions II: other

boundary conditions

In this chapter, we continue analyzing the sensitivity of mixing times to boundary

conditions in phase coexistence regimes, specifically when q is sufficiently large and

p = pc(q). As emphasized, the metastability of the FK Glauber and Swendsen–

Wang dynamics here on (Z/nZ)2 derives from the bottleneck between the ordered

and disordered phases. In Chapter 7, we saw that boundary conditions that pick out

the ordered or disordered phases substantially speed up mixing of the corresponding

FK and Potts models, and conjecturally lead to a picture governed by droplet

scaling and mean-curvature flow similar to the low-temperature Ising model. In this

section, we classify the mixing times under boundary conditions that are piecewise

ordered (i.e., wired or monochromatic—which we denote by R for red), disordered

(i.e., free), and periodic. We find that different combinations of these can lead to

262

either sub-exponential or exponential mixing depending on whether they preserve

or break the order-disorder phase symmetry.

We begin with some equilibrium estimates regarding interface fluctuations

in tilted domains in Section 8.0.1. Then, in Section 8.1, we prove Theorem 10,

showing sub-exponential mixing time upper bounds under Dobrushin boundary

conditions—those that induce a single macroscopic interface. In Section 8.2, we

prove Theorem 11, showing a similar upper bound on cylinders (c.f., torii where

mixing is exponentially slow). In Section 8.3, we show exponentially slow mixing

under mixed boundary conditions (e.g., wired on ∂n,sΛ and free on ∂e,wΛ) that

induce a pair of macroscopic interfaces (Theorem 9).

Finally, let us recall Remark 1.4.1, noting that some of these boundary conditions,

especially those on the cylinder, had not, to our knowledge, been studied in the

case of the low-temperature Ising Glauber dynamics: our results can be translated

to that setting (replacing wired and free by plus and minus) and the proofs would

only simplify.

8.0.1 Interface fluctuations on tilted subsets of Λn,n

In this section, we extend the surface tension estimate of Proposition 7.1.4 to

tilted half-infinite and finite strips. We first define subsets of Sn = J0, nK×J−∞,∞K

we consider in obtaining the desired sub-exponential upper bounds of Theorems 9–10

(see Figure 8.1).

Definition 8.0.1. For every n, which will be understood by context and omitted

from the following notation, define a tilted strip Sb,h,φ as (for b ∈ R, h ∈ R+,

263

φ

2h

(0, b)

Sb,h,φ

Sn

•

•

(0, b)

0

H+b,φ

H−b+a,φ ∩H+b,φ

a

•

•

Figure 8.1: An illustration of some of the sets and interfaces considered in Defini-tion 8.0.1 and Proposition 8.0.2. The (1, 0, φ) boundary conditions are depictedwith bold lines being wired and dashed lines being free.

φ ∈ (−π2, π

2)),

Sb,h,φ = (x, y) ∈ Sn : y ∈ Jb− h+ x tanφ, b+ h+ x tanφK .

Also, define the half-infinite strips,

H+b,φ = (x, y) ∈ Sn : y ≥ b+ x tanφ , H−b,φ = (x, y) ∈ Sn : y ≤ b+ x tanφ .

The following estimate—extending an estimate of [72], which in turn adapts [30]

to the FK model, to half-infinite strips—is a consequence of monotonicity of the

FK model and Proposition 7.1.4.

Proposition 8.0.2. Fix δ > 0 and let q be sufficiently large. For any b ∈ R and

φ ∈ [−π2

+ δ, π2− δ], consider the critical FK model on H+

b,φ ⊂ Sn with boundary

264

conditions (1, 0, b, φ) denoting wired boundary conditions on ∂H+b,φ ∩H

+b+1,φ and free

boundary elsewhere on ∂H+b,φ. There exist constants A > 0 and c(δ, q) > 0 such that

the order-disorder interface I, satisfies

π1,0,b,φ

H+b,φ

(I 6⊂ H−b+a,φ ∩H+b,φ) ≤ An2 exp

(−ca2/n

).

Proof. The proposition was proven in the case φ = 0 in Proposition 7.1.7 combining

monotonicity of the FK model in boundary conditions with Proposition 7.1.4. The

same proof carries over to the case φ 6= 0 as long as φ is uniformly bounded away

from ±π2

as the surface tension estimate of Proposition 7.1.6 on Sn is expressed in

that setup.

The following is the main equilibrium estimate we will use in the proofs of

sub-exponential mixing for general Dobrushin boundary conditions.

Proposition 8.0.3. Fix δ > 0 and let q be sufficiently large. For any b ∈ R,

h ≤ m ≤ n, and φ ∈ [−π2

+ δ, π2− δ], consider the critical FK model on Sb,m,φ∩Λn,n

with (1, 0, b, φ) boundary conditions denoting wired on ∂(Sb,m,φ ∩Λn,n)∩H+b+1,φ and

free elsewhere on ∂(Sb,m,φ ∩ Λn,n). Then there exists c(δ, q) > 0 such that

π1,0,b,φSb,m,φ∩Λn,n

(I 6⊂ Sb,h,φ ∩ Λn,n) . n2 exp(−ch2/n) .

We need the following preliminary estimate (see Figure 8.2 as a guide).

Lemma 8.0.4. Fix δ > 0 and let q be sufficiently large. For any b ∈ R and

φ ∈ [−π2

+ δ, π2− δ], consider the critical FK model on H+

b,φ ∩ Λn,n with (1, 0, b, φ)

boundary conditions denoting wired on ∂(H+b,φ ∩ Λn,n) ∩H+

b+1,φ and free elsewhere.

265

m h

b

Sb,h,φ

Sb,m,φ

•

•

b •

•φ

0

H+b,φ ∩ Λn,n

a

Figure 8.2: Left: the sets considered in Proposition 8.0.3, where we bound theprobability of the interface exceeding height ±h. Right: the proof of Lemma 8.0.4uses monotonicity in b.c., to go from bounding the probability of the order-disorderinterface exceeding height a in the gray set to the union of the gray and purplesets, to all shaded regions.

There exists c(δ, q) > 0 such that

π1,0,b,φ

H+b,φ∩Λn,n

(I 6⊂ H−b+a,φ ∩H+b,φ ∩ Λn,n) . n2 exp(−ca2/n) .

Proof. Let B = H−b+a,φ ∩H+b,φ ∩ Λn,n and consider the region S ′ = H+

b,φ ∩H−n,0.

For a general domain D ⊂ Sn let the boundary conditions (1, 0, b, φ) on it denote

wired on ∂D ∩ H+b+1,φ and free elsewhere on ∂D. By monotonicity in boundary

conditions, and then inclusion of events,

π1,0,b,φ

H+b,φ∩Λn,n

(I 6⊂ B) ≤ π1,0,b,φS′ (IΛn,n 6⊂ B)

≤ π1,0,b,φS′ (I 6⊂ H−b+a,φ ∩H

+b,φ) .

266

The application of the FKG inequality here is valid as under (1, 0, b, φ) boundary

conditions, events of the sort “I exceeds some height a”, are decreasing events, as

adding edges only pushes the order-disorder interface down. This property will be

used throughout the paper. By monotonicity in boundary conditions again,

π1,0,b,φS′ (I 6⊂ H−b+a,φ ∩H

+b,φ) ≤ π1,0,b,φ

H+b,φ

(I 6⊂ H−b+a,φ ∩H+b,φ) .

By Proposition 8.0.2, there exists a c(δ, q) > 0 such that

π1,0,b,φ

H+b,φ

(I 6⊂ H−b+a,φ ∩H+b,φ) . n2e−ca

2/n .

Proof of Proposition 8.0.3. As before, let (1, 0, b, φ) boundary conditions on

D ⊂ Z2 denote those that are wired on ∂D ∩H+b+1,φ and free on ∂D ∩H−b+1,φ. By

a union bound, write

π1,0,b,φSb,m,φ∩Λn,n

(I 6⊂ Sb,h,φ) ≤ π1,0,b,φSb,m,φ∩Λn,n

(I 6⊂ H+b−h,φ) + π1,0,b,φ

Sb,m,φ∩Λn,n(I 6⊂ H−b+h,φ)

(8.0.1)

and consider the two quantities independently. Observe that the first event on

the right-hand side is an increasing event, while the second event is a decreasing

event. By reflection symmetry and self-duality, if we prove the desired bound on the

latter, for general b, h, φ, it implies the former also. By monotonicity in boundary

conditions,

π1,0,b,φSb,m,φ∩Λn,n

(I 6⊂ H−b+h,φ) ≤ π1,0,b,φ

Sb,m,φ∩H+b−1,φ∩Λn,n

(I 6⊂ H−b+h,φ)

≤ π1,0,b,φ

H+b−1,φ∩Λn,n

(I 6⊂ H−b+h,φ) .

267

The right-hand side above is exactly the probability bounded in Lemma 8.0.4, from

which, along with the symmetry noted above and (8.0.1), the desired upper bound

follows.

We will also need the following bound in order to prove the mixing time upper

bounds on cylinders in §8.2. It is an adaptation of the proof of [70, Lemma A.6]

from the Ising model to the FK model via Propositions 7.1.4–7.1.5, and we omit

some details.

Proposition 8.0.5. Fix q to be large enough and ε > 0, and consider the critical

FK model on Λn,h for n12

+ε ≤ h ≤ n with 1, 0 boundary conditions denoting wired on

∂sΛn,h and free elsewhere. For ρ ∈ (0, 1) and δ small enough, there exists c(ε) > 0

such that

π1,0Λn,h

(I ∩ Jbn2c − δn, bn

2c+ δnK× J0, ρhK = ∅) & e−c(ρh)2/n .

Proof. Denote by B = Jbn2c − δn, bn

2c+ δnK× J0, ρhK. Recall the definition of the

cigar-shaped region Uκ,d,φ in (7.1.2). Following [70], for every − log2 n+ 2 ≤ i ≤

log2 n− 2, let zi be the nearest vertex to

(xi, d

∣∣∣∣ xi(n− xi)n

∣∣∣∣ 12

+κ)

where xi =n

2+

sgn i

4

|i|−1∑j=1

2−j .

Let Uzi,zi+1be the cigar shaped region zi+Uε/2,(1−ρ)∧ρ,φzi,zi+1

where d = (1+ρ)h/n12

+ε

and φzi,zi+1is the angle of the vector from zi to zi+1 (see, e.g., [70, Fig. 8]).

By monotonicity in boundary conditions and I ∩B = ∅ being an increasing

event,

π1,0Λn,h

(I ∩B = ∅) ≥ π1,0

H−h,0(I ∩B = ∅) ,

268

where (1, 0) boundary conditions on ∂H−h,0 are wired on ∂H−h,0 ∩ H−0,0 and free

elsewhere. Since Uzi,zi+1∩ B = ∅ for all i, if I ⊂

⋃|i|≤log2 n

Ui, then the desired

property holds.

Lemma A.6 of [70] gives a lower bound on the partition function restricted to

interfaces contained in⋃Ui as defined above in the setting of the Ising model;

the same proof extends that lower bound to the partition function of such FK

interfaces, noting that Proposition 7.1.5 is an analogue of [30, Theorem 4.16]: there

exists c > 0 such that

∑I⊂H−h,0

λ|I|+∑C∩I6=∅ Φ(C,I)1I ⊂

⋃Ui ≥ exp(−βcτ1,0(0)n− c(ρh)2/n) ,

where an error of (log n)O(1) was absorbed into the term c(ρh)2/n via the assumption

that h ≥ n1/2+ε and a choice of a suitable constant c > 0.

Furthermore, Proposition 7.1.5, in particular (7.1.4), implies there exists c > 0

so that

∑I⊂H−h,0

λ|I|+∑C∩I6=∅ Φ(C,I) ≤ exp(−βcτ1,0(0)n+ c(log n)c) .

Then writing the desired probability as in [70, (A.36)] as

π1,0

H−h,0(I ∩B 6= ∅) ≥ π1,0

H−h,0

(I ⊂

⋃|i|≤log2 n−2 Ui

)=

∑I⊂H−h,0

λ|I|+∑C∩I6=∅ Φ(C,I)1I ⊂

⋃Ui∑

I⊂H−h,0λ|I|+

∑C∩I6=∅ Φ(C,I)

,

and plugging in the above lower bound on the numerator and upper bound on the

denominator, concludes the proof.

269

8.0.2 Canonical paths estimate under typical FK boundary

conditions

In Section 6.4, we showed a canonical paths bound on n × l rectangles with

arbitrary realizable boundary conditions on three sides (but crucially not on one of

the long sides). In the proof of Theorems 10–11, we will require a similar bound

when both long-sides of Λn,l may have realizable boundary conditions carrying

long-distance information. We leverage the fact that we are at a discontinuous

phase transition point to prove a similar canonical paths estimate, but now only

for typical instances of these boundary conditions.

Proposition 8.0.6. Let q > 4, p = pc(q), and φ ∈ (−π2, π

2). Consider the FK

model on S = Sb,h,φ ∩ Λn,n with boundary conditions ξ ∼ P, where P is arbitrary

on ∂S ∩ ∂e,wΛn,n, and piecewise dominates wired-at-infinity or is dominated by

free-at-infinity on each of the other sides of ∂S. There exists Cq > 0 such that for

every sequence fn →∞, the FK Glauber dynamics have

P(ξ : tξmix ≥ |E(S)|2 exp[2(4h+ fn) log q]

)≤ O(n2e−Cqfn) .

The proof will use the following straightforward comparison estimate.

Lemma 8.0.7. Fix any p ∈ (0, 1) and q ≥ 1 and consider the FK model on

S = Sb,h,φ ∩ Λn,n with arbitrary boundary conditions ξ. For a connected edge subset

F ⊂ E(S), we have for every ω ∈ 0, 1E(S) and every ζ, ζ ′ ∈ 0, 1F

πξS(ω | ωF = ζ) ≤ q|∂F−∂S|+k(ξ,F )−1πξS(ω | ωF = ζ ′)

where k(ξ, F ) is the number of components of ξ that intersect both ∂F and ∂S−∂F .

270

Proof. By the domain Markov property, the difference between πξS(· | ωF = ζ) and

πξS(· | ωF = ζ ′) is in the boundary conditions that ζ ∪ ξ and ζ ′∪ ξ induce on S−F .

The boundary partitions induced can differ arbitrarily on ∂F − ∂S contributing

a factor of q|∂F−∂S|; on the other hand, since both configurations use ξ boundary

conditions on S, the boundary partitions on the rest of ∂(S − F ) can only differ if

ζ induces additional boundary connections between distinct boundary components

of ξ that reach ∂S − ∂F ; this accounts for the factor of qk(ξ,F )−1.

Proof of Proposition 8.0.6. We modify the proof of the canonical path estimate

for spin-systems with short-range interactions to the present setup. Partition every

segment L of ∂S \ ∂e,wΛn,n on which ξ is independently sampled, into b |L|fnc sub-

segments `i of fn vertices each, and possibly an additional exceptional segment `0.

Recall that for every edge e ∈ L, we defined Be to be the set of components of ξ

that contain vertices on both sides of L− e, called bridges : see Section 3.2. Let

Efn =

ξ :⋂L

⋂i≥1

∃e ∈ `i s.t. |Be| ≤ 1. (8.0.2)

We first prove that for every ξ ∈ fn, the FK Glauber dynamics on S has

gapξ,S ≤ |E(S)|(1 +√q) exp[2(4h+ fn) log q] . (8.0.3)

This will follow from a standard application of the canonical paths argument

for spin-systems with short-range interactions. Namely, label the edges in S

lexicographically in their midpoint, first by horizontal coordinate, then by vertical

coordinate, and define the path ζ(ω, ω′) (identified with a sequence of edges in Ωrc

between FK configurations, ω and ω′) as follows: let el1 , ..., elk be the sequence of

271

edges on which ω(eli) 6= ω′(eli), labeled in their lexicographic ordering. The i’th

edge in ζ(ω, ω′) will then be between configuration

η = ω′e1,...,eli−1 ∪ ωeli ,...,e|E(S)|

and its neighbor η′ which also has η′(ei) = ω′(ei). Also, let η∗ be the configuration

that is instead given by η∗ = ωe1,...,eli−1 ∪ ω′eli ,...,e|E(S)|. Then by Lemma 8.0.7

applied with the choice of F being e1, ..., eli−1 or eli , ..., e|E(S)|,

πξS(ω)πξS(ω′) ≤ πξS(η)πξS(η∗)q2(4h+fn)

as |∂F − ∂S| ≤ 2h and k(ξ, F ) ≤ 2h+ fn. This follows from the fact that ξ ∈ Efn

and the nested structure of boundary bridges, and the fact that the sides with

arbitrary boundary conditions have height at most 2h. By construction, for every

transition (η, η′), the map (ω, ω′) 7→ (η, η∗) is injective. Moreover, the probability of

making any transition in Ωrc is bounded below by 11+√q. Putting all this together,

by the path method we see that for every ξ ∈ Efn , Eq. (8.0.3) holds and by (2.2.3),

the corresponding bound with an extra factor of O(|E(S)|) also holds for the mixing

time.

It remains to bound the P-probability that ξ ∈ Efn . Fix a segment L of

∂S \ ∂e,wΛn,n on which P is piecewise sampled, and fix a sub-segment `i, then

take a union bound over all such segments and all `i. Moreover, suppose that

the boundary conditions on the segment L are dominated by free-at-infinity (the

estimate for the case when the distribution of P on ξL dominates wired-at-infinity

follows by similar reasoning). Since |Be| ≥ 1 is an increasing event, it suffices to

272

show

π0Z2(∃e ∈ `i s.t. |Be| = 0) ≥ 1−O(n2e−Cqfn) .

However, by planarity of boundary conditions induced by π0Z2 on ∂S, the complement

of the left-hand side is the event that there exist x, y in the two parts of L− `i such

that x←→ y, which, if Cq is the constant from (5.1.1), has probability at most

|L− `i|e−Cqfn ≤ 2ne−Cqfn .

(For the wired-at-infinity boundary conditions, observe that in order for |Be| > 1

for every e ∈ `i, there must exist x, y in the two parts of L− `i such that x∗←→ y:

this is in turn a decreasing event with exponentially decaying probability under

π1Z2 .) Taking a union bound over at most 4n sub-segments `i of various segments

L, we obtain that

P(ξ ∈ EC) ≤ 8n2e−Cqfn .

Remark 8.0.8. By standard comparison estimates, one could allow arbitrary

boundary conditions on any boundary segment of size O(h) of ∂S, paying a cost

in the spectral gap of at most exp(ch log q). This would follow from bounding the

ratio of the Dirichlet forms and the Radon-Nikodym derivative between the two

(see e.g., [66, Lemma 2.8] and [42, Eq. (5.1)] for details).

273

8.1 Subexponential mixing time with Dobrushin

boundary conditions

In this section, we consider the mixing time of FK Glauber and Swendsen–Wang

dynamics with boundary conditions that are free on a subset of ∂Λ and wired/red

elsewhere. While §8.3 demonstrates that such boundary conditions can induce a

slow mixing (at least exp(cn)) by respecting the order-disorder phase symmetry, this

section will establish that the mixing time is faster (at most exp(n1/2+o(1))) under

boundary conditions that have a single order-disorder interface. Define a general

class of order-disorder Dobrushin boundary conditions, whose FK representation is

wired on one connected boundary arc and free elsewhere. Let an, bn be two distinct

points on ∂Λn,n. For marked boundary points (an, bn), FK Dobrushin boundary

conditions are those that are wired on the clockwise (starting from the origin) arc

(an, bn) and free on (bn, an).

Sketch of proof

Our proof of Theorem 10 adapts the proof of tmix . exp(c√n log n) in [66] for

the low-temperature Ising model with plus boundary conditions, but using the

censoring inequality Theorem 2.2.7 instead of the block dynamics approach of [66].

For Dobrushin boundary conditions between (a, b), we sequentially (at times ti)

censor all updates except those in the strip Bi of height√n log n parallel to the

line segment 〈a, b〉 such that Bi and Bi+1 overlap on half their height. On the one

hand, the canonical paths estimate Proposition 8.0.6 bounds the mixing time of Bi

by exponential in its height, so that we take ti − ti−1 = O(ec√n logn). On the other

hand, by Theorem 2.2.7, if the censored chains started from all wired and all free are

274

coupled with high probability, that bounds the mixing time of the original chain. To

couple these two chains, we systematically push the interfaces of the chains started

from these initial configurations down (resp., up), until they are within O(√n log n)

of each other. This is possible because with high probability (see Lemma 8.0.4) the

interface of the bottom boundary of Bi (where the free initial configuration is seen)

never reaches the top of Bi−1 and the censored dynamics continues pushing the

interface down to O(√n log n) distance of 〈a, b〉 (see Figure 8.4).

Proof of Theorem 10. If an, bn are on the same side of ∂Λn,n, rotate Λn,n so that

they are both on ∂sΛn,n and the angle of the interface between them will be zero.

Then the proof below when an and bn are on different sides applies identically; the

only difference is the boundary conditions in the last step of recursion, where the

identity coupling still couples all FK chains with probability 1 in the mixing time

of that last block.

Now suppose an = (a1n, a

2n) and bn = (b1

n, b2n) are on different sides of ∂Λn,n; by

rotational and reflective symmetry and self-duality, we can take an ∈ ∂wΛn,n to

be the first point encountered clockwise from the origin, and ensure that φn =

tan−1(a2n−b2na1n−b1n

) is such that φn ∈ [−π4, π

4]. Fix any such choice of an, bn ∈ ∂Λn,n and

let φ = φn.

We establish the theorem for FK Glauber dynamics with (an, bn) Dobrushin

boundary conditions. Throughout this proof, let c1 = c1(q) > 0 be a large enough

constant (e.g., 5/Cq for Cq from Proposition 8.0.6 would suffice). Define the

275

overlapping blocks

Bi := San+(N−i)`,`,φ ∩ Λn,n (i = 1, . . . , N) and

Bi := San−(N−i)`,`,φ ∩ Λn,n (i = −1, . . . ,−N + 1) for N =⌈n`

⌉,

where we choose ` = c3

√n log n for c3 = 4/

√c2, with c2(q) as given by Proposi-

tion 8.0.2, so N ∼ c−13

√n/ log n (see Figure 8.3). Because of our choice of (B±j)

Nj=1,

as many as N of the Bj may be empty, and henceforth if Bj is empty, we say any

associated events hold trivially. Let

ti = i ·K log n ·N2 · |E(Bi ∪B−i)|2e4(4`+c1 logn) log q for 0 ≤ i ≤ N ,

for K to be chosen large later. Define the censored chain Xt: between times ti−1

and ti censor all updates except those in Bi ∪B−i. Let X1t be the censored chain

started from X0 = 1 and X0t be the censored chain started from X0 = 0. By

Theorem 2.2.7 and (2.2.7), it suffices to show that there exists a coupling of X0t

and X1t such that

P(X1tN6= X0

tN) = o(n−2) , (8.1.1)

since for any K, c1 fixed, we have tN . exp(O(√n log n)) as desired.

We now define a monotone coupling of X1t and X0

t which satisfies the above.

For each i = 1, . . . , N , define the event

A±i =X0tiE(R±i ) 6= X1

tiE(R±i )

, where R±i =

i⋃j=1

B±j \B±(i+1) .

276

an

bn

: Bi

: Bi+1

: R+i

BN

2`

2`

•

•

Figure 8.3: The blocks Bi in Λ, each of which are wired on all sides except possibly∂sBi and the region R+

i on which we have coupled.

We can then write under our coupling,

P(X0tN6= X1

tN) ≤ P

( N⋃i=1

(A+i ∪ A−i )

)≤ P(A+

N−1) + P(A−N−1) + P(AN | (A+N−1)c, (A−N−1)c) .

The bounds on P(A+N−1) and P(A−N−1) are analogous (using the duality of the FK

model) and therefore we only bound the former. Abusing notation slightly, when

we consider the restriction of the chain X1/0t to a boundary ∂S , we mean the

boundary conditions induced on that line by X1/0t Λ−S. We will prove the following

inductively.

Claim 8.1.1. There exists c(q) > 3 so that, for large enough K, c1, the following

holds.

(1) For every 1 ≤ i ≤ N−1, there exists an event Fi measurable w.r.t. (X0t , X

1t )t≤ti

such that P(Fi) ≥ 1−O(in−c) and P(X0ti∂nBi+1

∈ · | Fi) dominates the boundary

conditions induced by π1R+N−1

(·R+i

) on ∂nBi+1.

277

(2) For every 1 ≤ i ≤ N − 1,

P(A+i ) = P(X1

tiR+

i6= X0

tiR+

i) . in−c .

Proof of Claim 8.1.1. For the base case, (1) holds trivially because the outer

boundary on B1 is just given by the boundary of ∂Λ which will be all-wired there.

The base case proof of (2) is just a simplification of the proof of the inductive step

for (2) so we do not repeat it here. Now suppose both (1) and (2) hold for some

i− 1 and prove they hold for i for a c(q) > 0 we will pick later. By the inductive

hypothesis, with probability 1−O((i−1)n−c), we have X0tiR+

i−1= X1

tiR+

i−1, and the

boundary conditions on ∂nBi dominate wired-at-infinity. In particular, since on the

other sides of Bi both chains’ boundary conditions are either all-free or all-wired, by

Proposition 8.0.6, with probability 1−O((i− 1)n−c)−O(n−Cqc1+2) (where Cq > 0

is the constant from that proposition) the mixing time of FK Glauber dynamics

with boundary conditions given by X1ti∂Bi is at most |E(Bi∪B−i)|2e4(4`+c1 logn) log q.

Now suppose that both of these events hold and consider the probability that

X1tiR+

i6= X0

tiR+

i.

By submultiplicativity of total-variation distance, with probability 1−O(n−K)

there exists a coupling of X1/0ti Bi to π

X1/0ti−1

Bi. It remains to compute the probability

of (1) obtaining boundary conditions under πX0ti−1

Bithat dominate those induced by

π1R+N−1

(·R+i

) on ∂nBi+1, and (2) succeeding in coupling πX1ti−1

Bito π

X0ti−1

Bion Bi−Bi+1

(and therefore on all of R+i ).

Under the above events, let (ζ, 0) be the boundary conditions induced on ∂Bi by

X0ti−1

and (ζ, 1) be those induced by X1ti−1

where ζ is a random boundary condition

sampled from a distribution dominating π1R+N−1

(·R+i−1

) on ∂nBi. Then, the monotone

278

an

bn

1/0

BN

“1”

•

•

•

•

an

bn

1/0

BN

“1”

Figure 8.4: The green and blue blocks Bk, Bk+1 are updated by the censoreddynamics in two consecutive steps; with high probability, the chain (X0

tk)k pushes

its interface down toward 〈a, b〉 by ` at every step, and is subsequently coupled to(X1

tk)k on the growing gray region.

coupling of πζ,0Bi to πζ,1Bi couples the two on Bi − Bi+1 whenever the bottom-most

horizontal crossing I in the sample from πζ,0Bi has I ⊂ Bi+1 (see Figure 8.4). In

that case, by revealing dual-edges from the bottom up, the configurations from πζ,0Bi

and πζ,1Bi could be coupled above that interface and in particular on all of Bi−Bi+1.

(Observe that conditioning on the configuration below the interface, in order to

reveal I, cannot affect the boundary conditions above it because on each side

of ∂(R+i ∪ Bi+1) the boundary conditions are all-wired or all-free and additional

connections cannot be induced (cf. the boundary bridges of [43]).)

Observe, also, that when considering ζ dominating the boundary conditions

induced by π1R+N−1

(·R+i−1

), since the boundary on ∂Bi+1 ∩ ∂Λ is wired, by the

domain Markov property, the boundary conditions induced by X0ti+1

on ∂nBi+1

when I ⊂ Bi+1 holds will dominate π1R+N−1

(·R+i

). This implies part (1) if we can

bound πζ,0∂nBi(I ⊂ Bi+1).

Thus, for both (1) and (2) of the induction, it only remains to bound the

279

probability

E[πζ,0Bi(I 6⊂ Bi+1

) ∣∣ Fi−1

]≤ Eπ1

R+N−1

[πωR+i−1

,0

Bi(I 6⊂ Bi+1)

]≤ π1,0

R+i ∪Bi

(I 6⊂ Bi+1) , (8.1.2)

where (1, 0) boundary conditions on R+i ∪Bi are free on ∂sBi and wired elsewhere.

In that case, Lemma 8.0.4 (noting that the estimate there was independent of

b) with the choice of a = 12c3

√n log n implies that the probability in (8.1.2) is at

most O(n−6). Combining all of the above, the probability that items (1) and (2)

hold is at least

1−O((i− 1)n−c) +O(n−Cqc1+2) +O(n−K) +O(n−6) ,

which concludes the proof of the induction as long as we take c1, K large enough

that that the latter three terms are all o(n−c).

By Claim 8.1.1, we see that both P(A+N−1) and P(A−N−1) have probability at most

O(Nn−3) which is o(n−2). It remains to bound P(AN | (A+N−1)

c, (A−N−1)c) using

similar reasoning to the above. First of all, by part (2), with probability 1− o(n−2)

the chains X0tN−1

and X1tN−1

are coupled on both ∂nBN and ∂sBN . Moreover, by

part (1), the boundary conditions they induce dominates wired-at-infinity on ∂nBN

with probability at least 1− o(n−2) and likewise, are dominated by free-at-infinity

on ∂sBN with similar probability. Therefore, by time tN , by submultiplicativity of

total-variation distance and Proposition 8.0.6, we have

P(AN | (A+N−1)c, (A−N−1)c) ≤ 1− o(n−2)−O(n−Cqc1+2)−O(n−K) ,

280

and for c1, K large enough, the right-hand side is 1− o(n−2).

8.2 Sub-exponential mixing on cylinders

For the rectangle Λn,n define FK boundary conditions (p, 1) (resp. (p, 0) or

(p, 1, 0) boundary conditions) to be periodic boundary conditions on ∂n,sΛn,n and

wired boundary conditions on ∂e,wΛn,n (resp. free on ∂e,wΛn,n or wired on ∂wΛn,n

and free on ∂eΛn,n). We prove the mixing time upper bounds on cylinders with the

above boundary conditions (Theorem 11) at the same time. In what follows, we

use c > 0 to denote the existence of a constant (possibly depending on q), where

different appearances of c at different places may refer to different values.

The proof builds on the proof of Theorem 10 in that we use the censoring

inequalities to push the FK order-disorder across Λn,n in order to couple the

chains X1t and X0

t . We consider the censored dynamics that sequentially update

N = O(n12−ε) overlapping vertical strips of width n

12

+ε, ordered from left to right.

However, unlike the case in Theorem 10, our strips do not have wired boundary

conditions on three sides (their boundary conditions on the top and bottom are

periodic), and therefore, the interface is pushed to the next strip to be updated

with probability exp(−cn2ε) (rather than 1 − o(1)): see Figure 8.5. Thus, with

probability exp(−cn 12

+ε) the interface moves to next strip in N consecutive time

steps, so that one will succeed, with high probability, at pushing the interface

completely across Λn,n after exp(n12

+2ε) attempts.

Proof of Theorem 11. We again prove the upper bound for the FK Glauber

dynamics which translates to an upper bound on Swendsen–Wang dynamics by

Theorem 1.2.2. Fix any ε, δ > 0 small and consider blocks Bi for i = 1, ..., N where

281

N = dn`e − 1, given by

B2i−1 = J(i− 1)`, (i+ 1)`K× Jδn, (1− δ)nK

B2i = J(i− 1)`, (i+ 1)`K× J0, bn2c − δnK ∪ Jbn

2c+ δn, nK

where ` = n

12

+ε ,

and B2N+1 = B0 = Λ−∪Ni=1B2i−1 ∪B2i. Since the boundary conditions on ∂n,sΛn,n

are periodic, each B2i can be viewed as a single connected rectangle with boundary

∂n,sB2i = J(i− 1)`, (i+ 1)`K× bn2c+ δn, bn

2c − δn .

We prove the mixing time upper bound for (p, 1, 0) boundary conditions. We will

pause to comment where the (p, 1) boundary conditions would behave differently

(namely only when updating block B2N+1), and on why this does not affect the

proof. The (p, 0) boundary conditions can be treated by the dual version of the

argument we present.

We will cycle through the blocks Bi periodically, so define

Bj = Bj mod (2N+1) for all j > 2N .

Define the following censored Markov chain X1t (resp., X0

t ) started from initial

configuration 1 (resp., 0): for all i ≥ 0, Let fn = n12

+3ε and let

ti = i · n ·N2 · |E(Bi)|2e2(4`+fn) log q ;

during times [ti−1, ti), censor all updates outside block Bi. Let

T := t2N+1 exp(n12

+2ε) = exp(O(n12

+3ε)) ; (8.2.1)

282

by Theorem 2.2.7 and (2.2.7), it will suffice to show that

P(X1T 6= X0

T ) = o(n−2) .

We begin with a uniform upper bound on the mixing times of Bi.

Claim 8.2.1. Let m ≤ (2N + 1) exp(n12

+2ε) and, for every i ≤ m, define the event

Υi =(tX0ti−1

mix (Bi) ∨ tX1ti−1

mix (Bi))≤ |E(Bi)|2e2(4`+fn) log q

, (8.2.2)

where the superscript Xω0ti−1

denotes boundary conditions induced by Xω0ti−1

on Bi.

Then

P( ⋃i≤m

Υci

). m(n2e−Cqfn + e−n/2) , (8.2.3)

where Cq > 0 is the constant given by Proposition 8.0.6.

Proof of Claim 8.2.1. Let Ξi be the event that the law of the boundary conditions

on Bi under X0ti−1

piecewise dominate/are dominated by wired/free-at-infinity, resp.,

and likewise for X1ti−1

. We prove inductively that for every m ≤ (2N+1) exp(n12

+2ε),

P( ⋃i≤m

(Υci ∪ Ξc

i)

). m(n2e−Cqfn + e−n/2) . (8.2.4)

The base case, m = 1, has boundary conditions that are wired on ∂wB1 and

free/wired on ∂n,s,eB1, and thus Υ1 ∩ Ξ1 holds with probability 1 by a canonical

paths estimate (there are no distinct bridges). Suppose now that (8.2.4) holds for

some m; to show that it holds for m + 1, it suffices to show that the boundary

conditions induced by X0tm , i.e., the chain X0

t (the bound for the chain X1t follows

283

symmetrically).

Assume that the event⋂i≤m(Υi ∩ Ξi) holds. First of all, we notice that for any

i satisfying Υi, by the sub-multiplicativity of total-variation distance and definition

of ti,

‖P(X0tiBi ∈ ·)− π

X0ti−1

Bi‖tv ≤ e−n ; (8.2.5)

thus, a union bound over all i ≤ m = O(N exp(n12

+2ε)) implies that we may

construct a coupling of (X0t ) and some random variables Z1, . . . , Zm such that

P( ⋂i≤m

X0tiBi = Zi

)≥ 1− e−n/2 where Zi ∼ π

X0ti−1

Bifor each i .

We now claim that the boundary conditions induced by X0tm on Bm+1 are such

that they piecewise dominate/are dominated by wired/free-at-infinity on ∂Bm+1.

Consider the case wherem is even (the casem is odd follows analogously). According

to Ξm−1, if we denote by ζ the boundary conditions induced by X0tm−2

on ∂Bm−1,

then ζ piecewise dominates/is dominated by wired/free-at-infinity. Hence, when

sampling from πζBm−1, there would be a well-defined FK order-disorder interface I

between the boundary subsets that are alternately wired and free. For every such

interface, by the domain Markov property, the marginal under πζBm−1on each of

the connected components of E(Bm) \ I either dominates wired-at-infinity or is

dominated by free-at-infinity.

As a consequence, the boundary conditions on ∂n,s,wBm are piecewise sampled

from distributions dominating/dominated by wired/free-at-infinity, as are the

boundary conditions on the vertical bisector of (Bm−1 \Bm), a subset of ∂wBm+1.

Repeating this reasoning for the update on Bm, we see that the boundary conditions

284

on ∂n,s,wBm+1 are all sampled from distributions that piecewise dominate/are

dominated by wired/free-at-infinity. Finally, the same is true of ∂eBm+1 as it is

either completely free if m ≤ 2N + 1 or it is similarly sampled from πX0tj−1

Bjfor some

j < m which likewise satisfied Ξj. Thus we deduce that, except with probability

e−n/2, Ξm+1 holds. Then by Proposition 8.0.6, the boundary condition on Bm+1

induced by X0tm is such that

P(tX0tm

mix (Bm+1) ≥ |E(Bm+1)|2e2(4`+fn) log q) ≤ O(n2e−Cqfn) .

A union bound over the above errors concludes the proof.

Henceforth, we suppose that the event Υi holds for all i ≤ (2N + 1) exp(n12

+2ε),

which is the case with probability 1− exp(−Ω(n12

+3ε)) since fn = n12

+3ε.

Note that every time increment of t2N+1 we make an independent attempt

at coupling X1t to X0

t , albeit with initial configurations induced by the chains

at the end of the last sweep, and once the two chains are coupled on all of Λ,

they will remain coupled for all subsequent times. We will show that there exists

c(δ, q) > 0 such that for every k and every two configurations ω1k = X1

t(k−1)(2N+1)and

ω0k = X0

t(k−1)(2N+1),

P(X0tk(2N+1)

= X1tk(2N+1)

| ω1k, ω

0k)

≥ P(Ac2kN+k−1 | ω1k, ω

0k)P(Ack(2N+1) | Ac2kN+k−1, ω

1k, ω

0k)

& exp(−cn12

+ε) , (8.2.6)

285

where, in analogy to the proof of Theorem 10, if Ri =⋃j≤i mod (2N+1)Bj,

A2i =X0t2iR2i−B2i+1−B2i+2

6= X1t2iR2i−B2i+1−B2i+2

.

Eq. (8.2.6) is sufficient because the probability of not coupling X0T and X1

T by

time T (as defined in (8.2.1)) would then be bounded by

P(

Bin(exp(n12

+2ε), exp(−cn12

+ε)) = 0)

= o(n−2) .

In order to lower bound the probability in (8.2.6), we will construct a monotone

coupling of the two chains; therefore it suffices to consider the wired and free initial

configurations; by the Markov property, it also suffices to only consider the first

sweep k = 1.

Recall from (8.0.2) that for any rectangle Efn is the set of boundary conditions

on Bi such that in every boundary segment of length fn, there is an edge with

at most one boundary component containing vertices on both sides of that edge

(bridge).

Claim 8.2.2. There exists c(δ, q) > 0 such that, for large K and c1, the following

holds.

(1) For every 1 ≤ i ≤ N , there exists an event Fi measurable w.r.t. (X0t , X

1t )t≤t2i

such that P(X0t2i∂w(B2i+1∪B2i+2) ∈ · | Fi) is in Efn and dominates the wired-at-

infinity distribution on boundary conditions on ∂w(B2i+1 ∪B2i+2).

(2) For every 1 ≤ i ≤ N , the above defined event Fi satisfies

P(Fi, Ac2i) & exp(−cin2ε) .

286

Proof of Claim 8.2.2. For the base case of (1), observe that the boundary condi-

tions on ∂w(B1 ∪B2) dominate wired-at-infinity as they are all-wired; the base case

of (2) follows as the proof of the inductive step does, so we do not repeat it here.

Now assume that items (1) and (2) hold for i− 1 (for c(q) > 0 to be determined

later) and show that they hold for i for the same choice of c. Consider the middle

rectangle

Di = J(i− 1)`, (i+ 12)`K× Jbn

2c − δn, bn

2c+ δnK ,

and let I be the interface bounding the cluster(s) of X0tj∂wBj . Define the events

Γ2i =X0t2iB2i

: I ∩ (B2i −B2i+2) = ∅,

and

Γ2i−1 =X0t2i−1B2i−1

: I ∩Di = ∅∩ Γ2i−1 ,

where Γ2i−1 is the event that I is connected to e1, e2 in B2i−1−Di, where e1, e2 are

a pair of edges in either side of ∂wB2i−1− ∂wDi, with at most one bridge over them

(such a pair of edges exist since by assumption (1), the boundary conditions on

B2i−1 are in Efn). It is clear that both Γ2i−1 and Γ2i are increasing events. Then,

we can lower bound

P(Ac2i, Fi) ≥ P(Ac2i−2, Fi−1)P(Ac2i ∩ Fi ∩ Γ2i−1 ∩ Γ2i

∣∣ Ac2i−2, Fi−1

). (8.2.7)

By the inductive hypothesis, P(Ac2i−2, Fi−1) & exp(−c(i− 1)n2ε) and from now

on work in the probability space conditioned on Fi−1 ∩ Ac2i−2. Under Fi−1 ∩ Ac2i−2,

the boundary conditions X0t2i−2∂w(B2i−1∪B2i)

and X1t2i−2∂w(B2i−1∪B2i)

are coupled and

287

“1”“1”

“1”Di

0

0

B2i

“1”

e2

e1

Di

B2i−1

0

Figure 8.5: Left: the event Γ2i−1 where the interface of the component of ∂wB2i−1

does not intersect Di, and is connected to two edges e1, e2 that have no bridges.Right: the event Γ2i where the interface of the component of ∂wB2i∪∂n,sDi does notintersect the dashed line. Overall, the intersection Γ2i−1 ∩ Γ2i pushes the interfaceforward by `n.

dominate wired-at-infinity (let ζ denote that random boundary condition). In

the following time increment [t2i−2, t2i−1), only updates on B2i−1 are permitted;

since we are working on the event Υ2i−1, we can couple X0t2i−1B2i−1

to πζ,0B2i−1and

X1t2i−1B2i−1

to πζ,1B2i−1with probability 1−O(e−n) by (8.2.5).

Thus, by monotonicity, we consider the probability of the event I ∩Di = ∅ in

Γ2i−1 holding for a sample from πζ,0B2i−1. We claim that for some c(δ, q) > 0,

E[πζ,0B2i−1(I ∩Di = ∅) | Fi−1]

≥ Eπ1Z2

[πξ,0B2i−1(I ∩Di = ∅)]

& π1,0Λ(1−2δ)n,3`

(I ∩ Jn2− δn, n

2+ δnK× J0, 5`

2K = ∅)− e−c` ,

where the expectation is over all boundary conditions ξ induced by π1Z2 on ∂wB2i−1

288

and (1, 0) boundary conditions denote wired on ∂sΛ1,0(1−2δ)n,3`. Indeed, the second

inequality follows from considering the `-enlargement E`,2i−1 of B2i−1 which is

its concentric rectangle with extra side length `. If there is a wired circuit in

E`,2i−1 −B2i−1 under π1Z2 (by (5.1.1) this has probability 1− e−c`), we can replace

the expectation over b.c. induced by π1Z2 with an expectation over b.c. induced

by π1E`,2i−1

. Then extending the free boundary conditions on the other three sides

of B2i−1 all the way to ∂wE`,2i−1 and rotating yields the second inequality. By

Proposition 8.0.5 with the choices h = 3` and ρ = 56, there exists c(δ, q) > 0 so

that the probability in the right-hand side above is at least order e−cn2ε

(see e.g.,

Lemma 7.2.8 and Fig. 7.2 for a similar monotonicity argument).

Moreover, by the exponential decay of dual-connectivities, it is clear that for any

ζ ∈ Efn , we have πζ,0B2i−1(Γ2i−1) ≥ η for some η(q) > 0. Thus by the FKG inequality,

E[πζ,0B2i−1(Γ2i−1)] & e−cN

2ε

.

Let I2i−1 be the interface revealed by the component of ∂wB2i−1. Observe that

because Γ2i−1 is an increasing event, conditioned on Γ2i−1, if we reveal I2i−1 from

east to west, under the monotone coupling of πζ,1B2i−1to πζ,0B2i−1

the same edges would

also be open under πζ,1B2i−1; the same is also true of the edges that constitute Γ2i−1.

Having revealed these sets of open edges under both πζ,1B2i−1and πζ,0B2i−1

, by the

domain Markov property (there can not be distinct bridges over the interface we

have revealed), X1t2i−1Di = X0

t2i−1Di with probability at least (1−O(e−n))e−cn

2ε.

Now consider the next time increment [t2i−1, t2i) on B2i. Under the above

events, the configuration X0t2i−1Di π1

Z2(·Di), whence by (5.1.1), with probability

at least 1 − e−cδn, there is a pair of primal horizontal crossings of the top and

289

bottom halves of Di connecting ∂wB2i−1 to I2i−1. In that case, the distribution on

boundary conditions induced by X0t2i−1

(as well as X1t2i−1

) on ∂n,sB2i∩Di dominates

wired-at-infinity. Again, since we are working under the event Υ2i, we just consider

the event in Γ2i under πζ,0B2i. By applying (5.1.1) and enlarging the domains under

consideration as in the earlier bound on Γ2i−1, we obtain for some c(δ, q) > 0,

P(

Γ2i | Ac2i−2,Γ2i−1, Fi−1

)≥ (1−O(e−n))Eπ1

Z2

[πξ,0,(i+1/2)`B2i

(Γ2i)]

≥ (1−O(e−n))Eπ1Z2

[π0,ξ

Λ(1−2δ)n,3`/2(I ∩ Λ(1−2δ)n,` = ∅)

]≥ (1−O(e−n))(π0,1

Λn,2`(I ∩ Λn,3`/2 = ∅)− e−c`) .

Here, the boundary conditions (ξ, 0) in the first line denote ξ (over which we take

an expectation) induced on ∂B2i ∩ (x, y) : x = (i + 12)`, and free elsewhere on

∂B2i, and the boundary conditions in the second and third lines denote free on ∂n

of the boundary and respectively ξ and wired elsewhere. The second inequality is a

simple consequence of monotonicity in boundary conditions and the third inequality

follows from enlarging B2i by `/2 up to an error of e−c` coming from (5.1.1). By

Lemma 8.0.4 with φ = 0 and b = 1, there exists c(δ, q) > 0 such that the probability

on the right-hand side above is bounded below by (1−O(e−n))(1− e−cn2ε). In that

case, revealing the interface from east to west, we can couple X0t2i

to X1t2i

beyond

the interface (see also Fig. 8.5), so

P(Ac2i | A2i−2, Fi−1) ≥ (1−O(e−n))(1− e−cn2ε

)e−cn2ε

& exp(−cn2ε) . (8.2.8)

Finally, we claim that under the intersection of all the above events, with

probability 1−O(e−c`)−O(n2e−Cqfn), the boundary conditions induced by X1/0t2i

290

on ∂wB2i+1 and ∂wB2i+2 are in Efn and dominate wired-at-infinity, which combined

with (8.2.8) defines the desired set set Fi such that

P(Ac2i, Fi | A2i−2, Fi−1) & e−cn2ε

.

Recall that the configuration on Di under Γ2i−1 and Fi−1 dominates π1Z2Di . Then,

with probability 1 − 2e−cδn12 +ε

, Di contains two horizontal crossings connecting

∂wDi to I2i−1; since we are also conditioning on Γ2i, averaging over configurations

on Di, with probability 1− 3e−cδn12 +ε

, ∂wB2i+2 is surrounded by a wired circuit in

X0t2i

. Similarly, under X0t2i

, conditional on Γ2i−1, Γ2i and Fi−1, the configuration on

D′i = Ji`, (i+ 2)`K× J0, δnK ∪ J(1− δ)n, nK

below the interface revealed by Γ2i dominates π1Z2 so that with probability 1 −

2e−cδn12 +ε

, there are horizontal primal connections to that interface in both halves of

D′i. In that case, averaging over configurations in D′i with probability 1− 3e−cδn12 +ε

,

there is a wired circuit around ∂wB2i+1 in X0t2i

so that the distribution over boundary

conditions induced on ∂wB2i+1 also dominates wired-at-infinity. Moreover, as seen

in the proof of Proposition 8.0.6, a boundary condition dominating wired-at-infinity

is in Efn with probability 1−O(n2e−Cqfn). A union bound over the above concludes

the proof.

As a result, by item (2) of Claim 8.2.2, there exists some c(δ, q) > 0 for which

P(Ac2N) & exp(− cNn2ε

)& exp

(− cn

12

+ε).

Moreover, on that event, with high probability, the boundary conditions on ∂wB2N+1

291

induced by both X12N and X0

2N dominate wired-at-infinity. On the event Υ2N+1,

with probability 1−O(e−n) one can couple X1t2N+1

and X0t2N+1

to agree on B2N+1,

leading the two chains to be coupled on all of Λ. (It is only at this final step where

there is a difference between the (p, 1, 0) and (p, 1) boundary conditions; clearly,

if the coupling on B2N+1 succeeds in the former situation, it also succeeds in the

latter.)

8.3 Slow mixing with phase-symmetric bound-

ary conditions

Although the proof of slow mixing on the torus at a discontinuous phase

transition relied heavily on the topology of the torus (see proof of Theorem 2 in

[42]) we can—at least for sufficiently large q—also prove slow mixing in the presence

of mixed wired-free boundary conditions. Exploiting the self-duality, we see that

such boundary conditions still exhibit an exponential bottleneck, slowing down the

FK Glauber and Swendsen–Wang dynamics.

Definition 8.3.1. Let an, bn, cn, dn ∈ ∂Λn,n be a set of marked vertices ordered

clockwise from the origin around ∂Λn,n (by rotational symmetry, without loss of

generality assume an ∈ ∂wΛn,n). The mixed boundary conditions on (an, bn, cn, dn)

are those that are wired on the clockwise boundary arcs (an, bn) and (cn, dn) and free

on (bn, cn) and (dn, an)—all connected subsets of ∂Λn,n. We say that (an, bn, cn, dn)

are ε-separated if an ∈ ∂wΛn,n, at least one of bn, cn, dn is not contained in

J0, εnK× J0, nK, and

mini,j∈a,b,c,d; i 6=j

‖in − jn‖∞ ≥ εn .

292

For a choice of (an, bn, cn, dn), denote the FK measure with mixed boundary, πmixedΛ .

Remark 8.3.2. The requirement of ε-separation in our consideration of mixed

boundary conditions arises from the fact that if (an, bn, cn, dn) were all on ∂wΛn,n

repeating the proof of Theorem 10 with such boundary conditions would yield that

the mixing time is in fact sub-exponential. Clearly, if the four marked vertices

are sufficiently close to being on one side or to each other, a similar picture would

emerge. The requirement of macroscopic separation ensures that the bottleneck is

exponential in n.

With Definition 8.3.1 in hand, to prove Theorem 9, by rotational symmetry, we

wish to prove the following: let q be large, ε > 0, and consider the FK Glauber

dynamics for the critical Potts model on Λ = Λn,n with mixed boundary on

(an, bn, cn, dn) that are ε-separated. Then there exists c(ε, q) > 0 such that

tmix & exp(cn) .

Proof of Theorem 9. By (1.2.2) and Fact 2.2.6, it suffices to prove the bound

for the FK Glauber dynamics with mixed FK boundary conditions to deduce the

matching bound for Swendsen–Wang with Potts boundary conditions that are red

on the boundary arcs (an, bn) and (cn, dn) ⊂ ∂Λn,n and free elsewhere. Observe

that by planarity,

(an, bn)←→ (cn, dn)

=

(bn, cn)∗←→ (dn, an)

c,

293

and therefore, either

πmixedΛ

((an, bn)←→ (cn, dn)

)≤ 1

2, or πmixed

Λ

((bn, cn)

∗←→ (dn, an))≤ 1

2.

By self-duality of the class of ε-separated, mixed boundary conditions, we can

suppose without loss of generality that we are in the former case.

Recall the definition of the strips Sb,h,φ,H±b,φ in Definition 8.0.1, and let an =

(a1n, a

2n), and likewise for bn, cn, dn. Then let φa,d = tan−1(d

2n−a2

n

d1n−a1

n) and φb,c =

tan−1( c2n−b2nc1n−b1n

). Observe that by the ε-separation of (an, bn, cn, dn), one of φa,d and

φb,c is in [−π2

+ δ, π2− δ] for some small enough δ > 0 depending only on ε. Suppose

without loss of generality that for some δ(ε) > 0, φa,d ∈ [−π2

+ δ, π2− δ] and consider

the strip

S = H+a1n,φa,d

∩H−a1n+ε2n,φa,d

∩ Λ ,

Geometrically, by definition of ε-separation, S satisfies S ∩ ∂Λ ⊂ (an, bn) ∪ (cn, dn).

There is some x, h, φ = φa,d such that S = Sx,h,φa,d ∩ Λ; fix that x ∈ R+, h = ε2/2.

Define ∂nS = S ∩H+x+h−1,φ and ∂sS = S ∩H−x−h+1,φ, and let

A =

(an, bn)S←→ (cn, dn)

be the bottleneck set whose conductance Q(A,Ac)/(π(A)π(Ac)) we bound. Since

A ⊂ (an, bn)←→ (cn, dn) ,

294

we have that πmixedΛ (Ac) > 1

2. Therefore, we can write

gaprc ≤ 2Φ? ≤2Q(A,Ac)

πmixedΛ (A)πmixed

Λ (Ac)≤ 4πmixed

Λ (∂A | A) ,

(where ∂A := ω : P (ω,Ac) > 0 and we used a worst-case bound of 1 on the

transition rates in Q(A,Ac)), in which case it suffices to prove that for some

c(q) > 0,

πmixedΛ (∂A | A) . exp(−cε4n) .

For ω ∈ A, in order for P (ω,Ac) to be positive (ω ∈ ∂A), there must exist an

edge e in S that is pivotal to A, i.e., ω(e) = 1 and ω′ = ω − e /∈ A. We estimate

the probability πmixedΛ (∂A | A) by taking a union bound over the probability of any

edge, e, in E(S) being pivotal to S.

First examine whether e is closer in its y coordinate to ∂nS or ∂sS. Suppose

without loss of generality, we are in the former case, whence we expose the north-

most primal crossing of S, under πmixedΛ (ω | A) (revealing, first, the configuration

on Λ ∩H+x+h−1,φ, then the dual-components of ∂nS in S). Such a crossing exists by

conditioning on A.

Denote by ζ the horizontal crossing we have revealed as such. By the conditioning

on S, it is clear that ζ must connect (an, bn) to (cn, dn) in S. In order for e to be

pivotal to S, e must be an open edge in ζ and there must exist a dual crossing

connecting e to ∂sS. Let D be the southern connected component of E(Λ)− ζ; we

wish to bound

πmixedΛ

(e

D∗←→ ∂sS | A, ζ, ωζ = 1).

295

By monotonicity in boundary conditions, if we let R = D ∪ S, for every such ζ,

πmixedΛ (ωD | A, ζ, ωζ = 1) = π1,0

D π1,0R (ωD) ,

where (1, 0) boundary conditions denote free on ∂R− S and wired elsewhere.

We can decompose the probability

π1,0R

(e

D∗←→ ∂sS)≤ π1,0

R

(e

∗←→ ∂sS)

into the event Γ1 that the dual-component of ∂sS (and thus the interface of R with

(1, 0) boundary conditions) is a subset of S ∩H−x−h/2,φ, and Γc1. Under Γc1, since e

is closer in its y-coordinate to ∂nS, the vertical distance between e and ∂sS is at

least h/2 so that e /∈ S ∩H−x−h/2,φ and e cannot be dual-connected to ∂sS.

Bounding the probability of Γ1 by monotonicity in boundary conditions and

Proposition 8.0.3, there exists c(q) > 0 such that for every ζ,

πmixedΛ (Γ1 | A, ζ, ωζ = 1) ≤ π

1,0,a1n,φ

Sa1n,h/2,φ

∩Λ(I 6⊂ Sa1n,h/4,φ

∩ Λ) . n2 exp(−ch2n) .

Under Γ1 we can take a worst case bound of one on the probability of eD∗←→ ∂sS.

Therefore, for some c(q) > 0, we have πmixedΛ (∂A | A) . exp(−cε4n).

Using the above as a bound on Q(A,Ac)/πmixedΛ (A) in Eq. (2.2.14) and plugging

into Eq. (2.2.14) implies gap−1rc & exp(cε4n) for the FK Glauber dynamics with

mixed order-disorder boundary conditions.

296

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