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Clay Mathematics ProceedingsVolume 15, 2012
Conformal Invariance of Lattice Models
Hugo Duminil-Copin and Stanislav Smirnov
Abstract. These lecture notes provide an (almost) self-contained account on con-formal invariance of the planar critical Ising and FK-Ising models. They present thetheory of discrete holomorphic functions and its applications to planar statisticalphysics (more precisely to the convergence of fermionic observables). Convergenceto SLE is discussed briefly. Many open questions are included.
Contents
1. Introduction 2141.1. Organization of the notes 2181.2. Notations 2192. Two-dimensional Ising model 2202.1. Boundary conditions, infinite-volume measures and phase transition 2202.2. Low and high temperature expansions of the Ising model 2222.3. Spin-Dobrushin domain, fermionic observable and results on the Ising
model 2243. Two-dimensional FK-Ising model 2273.1. FK percolation 2273.2. FK-Ising model and Edwards-Sokal coupling 2293.3. Loop representation of the FK-Ising model and fermionic observable 2314. Discrete complex analysis on graphs 2354.1. Preharmonic functions 2354.2. Preholomorphic functions 2404.3. Isaacsâs definition of preholomorphic functions 2404.4. s-holomorphic functions 2414.5. Isoradial graphs and circle packings 2445. Convergence of fermionic observables 2455.1. Convergence of the FK fermionic observable 2455.2. Convergence of the spin fermionic observable 2506. Convergence to chordal SLE(3) and chordal SLE(16/3) 2526.1. Tightness of interfaces for the FK-Ising model 2536.2. sub-sequential limits of FK-Ising interfaces are Loewner chains 2546.3. Convergence of FK-Ising interfaces to SLE(16/3) 2566.4. Convergence to SLE(3) for spin Ising interfaces 2577. Other results on the Ising and FK-Ising models 2587.1. Massive harmonicity away from criticality 258
2010 Mathematics Subject Classification. Primary: 60K35; Secondary: 60J67, 82B23.
© 2012 Hugo Duminil-Copin and Stanislav Smirnov
213
214 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
7.2. Russo-Seymour-Welsh Theorem for FK-Ising 2627.3. Discrete singularities and energy density of the Ising model 2648. Many questions and a few answers 2658.1. Universality of the Ising model 2658.2. Full scaling limit of critical Ising model 2668.3. FK percolation for general cluster-weight q â„ 0 2668.4. O(n) models on the hexagonal lattice 2688.5. Discrete observables in other models 270References 272
1. Introduction
The celebrated Lenz-Ising model is one of the simplest models of statistical physicsexhibiting an order-disorder transition. It was introduced by Lenz in [Len20] as anattempt to explain Curieâs temperature for ferromagnets. In the model, iron is modeledas a collection of atoms with fixed positions on a crystalline lattice. Each atom has amagnetic spin, pointing in one of two possible directions. We will set the spin to be equalto 1 or â1. Each configuration of spins has an intrinsic energy, which takes into accountthe fact that neighboring sites prefer to be aligned (meaning that they have the samespin), exactly like magnets tend to attract or repel each other. Fix a box Î â Z2 of sizen. Let Ï â â1,1Î be a configuration of spins 1 or â1. The energy of the configurationÏ is given by the Hamiltonian
EÎ(Ï) â¶= â âxâŒy
ÏxÏy
where x ⌠y means that x and y are neighbors in Î. The energy is, up to an additiveconstant, twice the number of disagreeing neighbors. Following a fundamental principleof physics, the spin-configuration is sampled proportionally to its Boltzmann weight: atan inverse-temperature ÎČ, the probability ”ÎČ,Î of a configuration Ï satisfies
”ÎČ,Î(Ï) â¶= eâÎČEÎ(Ï)
ZÎČ,Î
where
ZÎČ,Î â¶= âÏââ1,1Î
eâÎČEÎ(Ï)
is the so-called partition function defined in such a way that the sum of the weights overall possible configurations equals 1. Above a certain critical inverse-temperature ÎČc,the model has a spontaneous magnetization while below ÎČc does not (this phenomenonwill be described in more detail in the next section). When ÎČc lies strictly between 0and â, the Ising model is said to undergo a phase transition between an ordered anda disordered phase. The fundamental question is to study the phase transition betweenthe two regimes.
Lenzâs student Ising proved the absence of phase transition in dimension one (mean-ing ÎČc = â) in his PhD thesis [Isi25], wrongly conjecturing the same picture inhigher dimensions. This belief was widely shared, and motivated Heisenberg to in-troduce his famous model [Hei28]. However, some years later Peierls [Pei36] usedestimates on the length of interfaces between spin clusters to disprove the conjecture,showing a phase transition in the two-dimensional case. Later, Kramers and Wannier[KW41a, KW41b] derived nonrigorously the value of the critical temperature.
CONFORMAL INVARIANCE OF LATTICE MODELS 215
Figure 1. Ising configurations at ÎČ < ÎČc, at ÎČ = ÎČc, and ÎČ > ÎČc respectively.
In 1944, Onsager [Ons44] computed the partition function of the model, followedby further computations with Kaufman, see [KO50] for instance1. In the physicalapproach to statistical models, the computation of the partition function is the firststep towards a deep understanding of the model, enabling for instance the computationof the free energy. The formula provided by Onsager led to an explosion in the number ofresults on the 2D Ising model (papers published on the Ising model can now be countedin the thousands). Among the most noteworthy results, Yang derived rigorously thespontaneous magnetization [Yan52] (the result was derived nonrigorously by Onsagerhimself). McCoy and Wu [MW73] computed many important quantities of the Isingmodel, including several critical exponents, culminating with the derivation of two-pointcorrelations between sites (0,0) and (n,n) in the whole plane. See the more recent bookof Palmer for an exposition of these and other results [Pal07].
The computation of the partition function was accomplished later by several othermethods and the model became the most prominent example of an exactly solvablemodel. The most classical techniques include the transfer-matrices technique devel-oped by Lieb and Baxter [Lie67, Bax89], the Pfaffian method, initiated by Fisherand Kasteleyn, using a connection with dimer models [Fis66, Kas61], and the com-binatorial approach to the Ising model, initiated by Kac and Ward [KW52] and thendeveloped by Sherman [She60] and Vdovichenko [Vdo65]; see also the more recent[DZM+99, Cim10].
Despite the number of results that can be obtained using the partition function,the impossibility of computing it explicitly enough in finite volume made the geomet-ric study of the model very hard to perform while using the classical methods. Thelack of understanding of the geometric nature of the model remained mathematicallyunsatisfying for years.
The arrival of the renormalization group formalism (see [Fis98] for a histori-cal exposition) led to a better physical and geometrical understanding, albeit mostlynon-rigorous. It suggests that the block-spin renormalization transformation (coarse-graining, e.g. replacing a block of neighboring sites by one site having a spin equal tothe dominant spin in the block) corresponds to appropriately changing the scale and the
1This result represented a shock for the community: it was the first mathematical evidence thatthe mean-field behavior was inaccurate in low dimensions.
216 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
temperature of the model. The Kramers-Wannier critical point then arises as the fixedpoint of the renormalization transformations. In particular, under simple rescaling theIsing model at the critical temperature should converge to a scaling limit, a continuousversion of the originally discrete Ising model, corresponding to a quantum field the-ory. This leads to the idea of universality: the Ising models on different regular latticesor even more general planar graphs belong to the same renormalization space, with aunique critical point, and so at criticality the scaling limit and the scaling dimensionsof the Ising model should always be the same (it should be independent of the latticewhereas the critical temperature depends on it).
Being unique, the scaling limit at the critical point must satisfy translation, rota-tion and scale invariance, which allows one to deduce some information about correla-tions [PP66, Kad66]. In seminal papers [BPZ84b, BPZ84a], Belavin, Polyakov andZamolodchikov suggested a much stronger invariance of the model. Since the scaling-limit quantum field theory is a local field, it should be invariant by any map which islocally a composition of translation, rotation and homothety. Thus it becomes naturalto postulate full conformal invariance (under all conformal transformations2 of subre-gions). This prediction generated an explosion of activity in conformal field theory,allowing nonrigorous explanations of many phenomena; see [ISZ88] for a collection ofthe original papers of the subject.
To summarize, Conformal Field Theory asserts that the Ising model admits a scalinglimit at criticality, and that this scaling limit is a conformally invariant object. Froma mathematical perspective, this notion of conformal invariance of a model is ill-posed,since the meaning of scaling limit is not even clear. The following solution to thisproblem can be implemented: the scaling limit of the model could simply retain theinformation given by interfaces only. There is no reason why all the information of amodel should be encoded into information on interfaces, yet one can hope that most ofthe relevant quantities can be recovered from it. The advantage of this approach is thatthere exists a mathematical setting for families of continuous curves.
In the Ising model, there is a canonical way to isolate macroscopic interfaces. Con-sider a simply-connected domain Ω with two points a and b on the boundary and ap-proximate it by a discrete graph ΩΎ â ÎŽZ
2. The boundary of ΩΎ determines two arcsâab and âba and we can fix the spins to be +1 on the arc âab and â1 on the arc âba(this is called Dobrushin boundary conditions). In this case, there exists an interface3
separating +1 and â1 going from a to b and the prediction of Conformal Field Theorythen translates into the following predictions for models: interfaces in ΩΎ converge whenÎŽ goes to 0 to a random continuous non-selfcrossing curve Îł(Ω,a,b) between a and b in Ω
which is conformally invariant in the following way:
For any (Ω, a, b) and any conformal map Ï â¶ Î© â C, the random curve Ï Îł(Ω,a,b)has the same law as Îł(Ï(Ω),Ï(a),Ï(b)).
In 1999, Schramm proposed a natural candidate for the possible conformally invari-ant families of continuous non-selfcrossing curves. He noticed that interfaces of modelsfurther satisfy the domain Markov property, which, together with the assumption of con-formal invariance, determine the possible families of curves. In [Sch00], he introducedthe Schramm-Loewner Evolution (SLE for short): for Îș > 0, the SLE(Îș) is the ran-dom Loewner Evolution with driving process
âÎșBt, where (Bt) is a standard Brownian
motion (see Beffaraâs course in this volume). In our case, it implies that the randomcontinuous curve Îł(Ω,a,b) described previously should be an SLE.
2i.e. one-to-one holomorphic maps.3In fact the interface is not unique. In order to solve this issue, consider the closest interface to
âab.
CONFORMAL INVARIANCE OF LATTICE MODELS 217
Figure 2. An interface between + and â in the Ising model.
Proving convergence of interfaces to an SLE is fundamental. Indeed, SLE processesare now well-understood and their path properties can be related to fractal propertiesof the critical phase. Critical exponents can then be deduced from these properties viathe so-called scaling relations. These notes provide an (almost) self-contained proof ofconvergence to SLE for the two-dimensional Ising model and its random-cluster repre-sentation the FK-Ising model (see Section 3 for a formal definition).
Main result 1 (Theorem 2.10) The law of interfaces of the critical Ising model convergesin the scaling limit to a conformally invariant limit described by the Schramm-LoewnerEvolution of parameter Îș = 3.Main result 2 (Theorem 3.13) The law of interfaces of the critical FK-Ising modelconverges in the scaling limit to a conformally invariant limit described by the Schramm-Loewner Evolution of parameter Îș = 16/3.
Even though we now have a mathematical framework for conformal invariance, itremains difficult to prove convergence of interfaces to SLEs. Observe that working withinterfaces offers a further simplification: properties of these interfaces should also beconformally invariant. Therefore, one could simply look at a discrete observable of themodel and try to prove that it converges in the scaling limit to a conformally covariantobject. Of course, it is not clear that this observable would tell us anything about criticalexponents, yet it already represents a significant step toward conformal invariance.
In 1994, Langlands, Pouliot and Saint-Aubin [LPSA94] published a number ofnumerical values in favor of conformal invariance (in the scaling limit) of crossing prob-abilities in the percolation model. More precisely, they checked that, taking differenttopological rectangles, the probability CÎŽ(Ω,A,B,C,D) of having a path of adjacentopen edges from AB to CD converges when ÎŽ goes to 0 towards a limit which is thesame for (Ω,A,B,C,D) and (ΩâČ,AâČ,BâČ,CâČ,DâČ) if they are images of each other by aconformal map. The paper [LPSA94], while only numerical, attracted many mathe-maticians to the domain. The same year, Cardy [Car92] proposed an explicit formula forthe limit of percolation crossing probabilities. In 2001, Smirnov proved Cardyâs formularigorously for critical site percolation on the triangular lattice [Smi01], hence rigorouslyproviding a concrete example of a conformally invariant property of the model. A some-what incredible consequence of this theorem is that the mechanism can be reversed:even though Cardyâs formula seems much weaker than convergence to SLE, they are
218 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
actually equivalent. In other words, conformal covariance of one well-chosen observableof the model can be sufficient to prove conformal invariance of interfaces.
It is also possible to find an observable with this property in the Ising case (seeDefinition 2.9). This observable, called the fermionic observable, is defined in termsof the so-called high temperature expansion of the Ising model. Specific combinatorialproperties of the Ising model translate into local relations for the fermionic observable.In particular, the observable can be proved to converge when taking the scaling limit.This convergence result (Theorem 2.11) is the main step in the proof of conformalinvariance. Similarly, a fermionic observable can be defined in the FK-Ising case, andits convergence implies the convergence of interfaces.
Archetypical examples of conformally covariant objects are holomorphic solutionsto boundary value problems such as Dirichlet or Riemann problems. It becomes naturalto expect that discrete observables which are conformally covariant in the scaling limitare naturally preharmonic or preholomorphic functions, i.e. relevant discretizations ofharmonic and holomorphic functions. Therefore, the proofs of conformal invariance har-ness discrete complex analysis in a substantial way. The use of discrete holomorphicityappeared first in the case of dimers [Ken00] and has been extended to several statisticalphysics models since then. Other than being interesting in themselves, preholomorphicfunctions have found several applications in geometry, analysis, combinatorics, and prob-ability. We refer the interested reader to the expositions by LovĂĄsz [Lov04], Stephenson[Ste05], Mercat [Mer01], Bobenko and Suris [BS08]. Let us finish by mentioning thatthe previous discussion sheds a new light on both approaches described above: combina-torial properties of the discrete Ising model allow us to prove the convergence of discreteobservables to conformally covariant objects. In other words, exact integrability andConformal Field Theory are connected via the proof of the conformal invariance of theIsing model.
Acknowledgments These notes are based on a course on conformal invariance of latticemodels given in BĂșzios, Brazil, in August 2010, as part of the Clay Mathematics InstituteSummer School. The course consisted of six lectures by the second author. The authorswish to thank the organisers of both the Clay Mathematics Institute Summer Schooland the XIV Brazilian Probability School for this milestone event. We are particularlygrateful to Vladas Sidoravicius for his incredible energy and the constant effort putinto the organization of the school. We thank StĂ©phane Benoist, David Cimasoni andAlan Hammond for a careful reading of previous versions of this manuscript. The twoauthors were supported by the EU Marie-Curie RTN CODY, the ERC AG CONFRA,as well as by the Swiss FNS. The research of the second author is supported by theChebyshev Laboratory (Department of Mathematics and Mechanics, St. PetersburgState University) under RF Government grant 11.G34.31.0026.
1.1. Organization of the notes. Section 2 presents the necessary background onthe spin Ising model. In the first subsection, we recall general facts on the Ising model.In the second subsection, we introduce the low and high temperature expansions, aswell as Kramers-Wannier duality. In the last subsection, we use the high-temperatureexpansion in spin Dobrushin domains to define the spin fermionic observable. Via theKramers-Wannier duality, we explain how it relates to interfaces of the Ising model atcriticality and we state the conformal invariance result for Ising.
Section 3 introduces the FK-Ising model. We start by defining general FK per-colation models and we discuss planar duality. Then, we explain the Edwards-Sokalcoupling, an important tool relating the spin Ising and FK-Ising models. Finally, weintroduce the loop representation of the FK-Ising model in FK Dobrushin domains. Itallows us to define the FK fermionic observable and to state the conformal invarianceresult for the FK-Ising model.
CONFORMAL INVARIANCE OF LATTICE MODELS 219
Section 4 is a brief survey of discrete complex analysis. We first deal with prehar-monic functions and a few of their elementary properties. These properties will be usedin Section 6. In the second subsection, we present a brief historic of preholomorphicfunctions. The third subsection is the most important, it contains the definition andseveral properties of s-holomorphic (or spin-holomorphic) functions. This notion is cru-cial in the proof of conformal invariance: the fermionic observables will be proved tobe s-holomorphic, a fact which implies their convergence in the scaling limit. We alsoinclude a brief discussion on complex analysis on general graphs.
Section 5 is devoted to the convergence of the fermionic observables. First, we showthat the FK fermionic observable is s-holomorphic and that it converges in the scalinglimit. Second, we deal with the spin fermionic observable. We prove its s-holomorphicityand sketch the proof of its convergence.
Section 6 shows how to harness the convergence of fermionic observables in orderto prove conformal invariance of interfaces in the spin and FK-Ising models. It mostlyrelies on tightness results and certain properties of Loewner chains.
Section 7 is intended to present several other applications of the fermionic ob-servables. In particular, we provide an elementary derivation of the critical inverse-temperature.
Section 8 contains a discussion on generalizations of this approach to lattice models.It includes a subsection on the Ising model on general planar graphs. It also gathersconjectures regarding models more general than the Ising model.
1.2. Notations.1.2.1. Primal, dual and medial graphs. We mostly consider the (rotated) square
lattice L with vertex set eiÏ/4Z2 and edges between nearest neighbors. An edge withend-points x and y will be denoted by [xy]. If there exists an edge e such that e = [xy],we write x ⌠y. Finite graphs G will always be subgraphs of L and will be called primalgraphs. The boundary of G, denoted by âG, will be the set of sites of G with fewerthan four neighbors in G.
The dual graph Gâ of a planar graph G is defined as follows: sites of Gâ correspondto faces of G (for convenience, the infinite face will not correspond to a dual site), edgesof Gâ connect sites corresponding to two adjacent faces of G. The dual lattice of L isdenoted by L
â.The medial lattice L
is the graph with vertex set being the centers of edges of L,and edges connecting nearest vertices, see Fig. 6. The medial graph G is the subgraphof L composed of all the vertices of L corresponding to edges of G. Note that L
isa rotated and rescaled (by a factor 1/â2) version of L, and that it is the usual squarelattice. We will often use the connection between the faces of L and the sites of L andLâ. We say that a face of the medial lattice is black if it corresponds to a vertex of L,
and white otherwise. Edges of L are oriented counterclockwise around black faces.
1.2.2. Approximations of domains. We will be interested in finer and finer graphsapproximating continuous domains. For ÎŽ > 0, the square lattice
â2ÎŽL of mesh-sizeâ
2ÎŽ will be denoted by LÎŽ. The definitions of dual and medial lattices extend to thiscontext. Note that the medial lattice L
ÎŽ has mesh-size ÎŽ.
For a simply connected domain Ω in the plane, we set ΩΎ = Ω ⩠LΎ. The edgesconnecting sites of ΩΎ are those included in Ω. The graph ΩΎ should be thought of asa discretization of Ω (we avoid technicalities concerning the regularity of the domain).More generally, when no continuous domain Ω is specified, ΩΎ stands for a finite simplyconnected (meaning that the complement is connected) subgraph of LΎ.
We will be considering sequences of functions on ΩΎ for Ύ going to 0. In order tomake functions live in the same space, we implicitly perform the following operation:for a function f on ΩΎ, we choose for each square a diagonal and extend the function to
220 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
Ω in a piecewise linear way on every triangle (any reasonable way would do). Since noconfusion will be possible, we denote the extension by f as well.
1.2.3. Distances and convergence. Points in the plane will be denoted by their com-plex coordinates, Re(z) and Im(z) will be the real and imaginary parts of z respectively.The norm will be the usual complex modulus ⣠â âŁ. Unless otherwise stated, distancesbetween points (even if they belong to a graph) are distances in the plane. The distancebetween a point z and a closed set F is defined by
d(z,F ) â¶= infyâFâŁz â yâŁ.(1.1)
Convergence of random parametrized curves (say with time-parameter in [0,1]) is inthe sense of the weak topology inherited from the following distance on curves:
d(Îł1, Îł2) = infÏ
supuâ[0,1]
âŁÎł1(u) â Îł2(Ï(u))âŁ,(1.2)
where the infimum is taken over all reparametrizations (i.e. strictly increasing continu-ous functions Ïⶠ[0,1]â [0,1] with Ï(0) = 0 and Ï(1) = 1).
2. Two-dimensional Ising model
2.1. Boundary conditions, infinite-volume measures and phase transition.The (spin) Ising model can be defined on any graph. However, we will restrict ourselvesto the (rotated) square lattice. Let G be a finite subgraph of L, and b â â1,+1âG.The Ising model with boundary conditions b is a random assignment of spins â1,+1(or simply â/+) to vertices of G such that Ïx = bx on âG, where Ïx denotes the spin atsite x. The partition function of the model is denoted by
ZbÎČ,G = âÏââ1,1G ⶠÏ=b on âG
expâĄâąâąâąâŁÎČ âxâŒy ÏxÏy
â€â„â„â„⊠,(2.1)
where ÎČ is the inverse-temperature of the model and the second summation is over allpairs of neighboring sites x, y in G. The probability of a configuration Ï is then equalto
”bÎČ,G(Ï) = 1
ZbÎČ,G
expâĄâąâąâąâŁÎČ âxâŒy ÏxÏy
â€â„â„â„⊠.(2.2)
Equivalently, one can define the Ising model without boundary conditions, alsocalled free boundary conditions (it is the one defined in the introduction). The measure
with free boundary conditions is denoted by ”fÎČ,G
.
We will not offer a complete exposition on the Ising model and we rather focus oncrucial properties. The following result belongs to the folklore (see [FKG71] for theoriginal paper). An event is called increasing if it is preserved by switching some spinsfrom â to +.
Theorem 2.1 (Positive association at every temperature). The Ising model on afinite graph G at temperature ÎČ > 0 satisfies the following properties:
FKG inequality: For any boundary conditions b and any increasing eventsA,B,
”bÎČ,G(A â©B) ℠”bÎČ,G(A)”bÎČ,G(B).(2.3)
Comparison between boundary conditions: For boundary conditions b1 â€b2 (meaning that spins + in b1 are also + in b2) and an increasing event A,
”b1ÎČ,G(A) †”b2ÎČ,G(A).(2.4)
CONFORMAL INVARIANCE OF LATTICE MODELS 221
If (2.4) is satisfied for every increasing event, we say that ”b2ÎČ,G
stochastically dom-
inates ”b1ÎČ,G (denoted by ”b1ÎČ,G †”b2ÎČ,G). Two boundary conditions are extremal for the
stochastic ordering: the measure with all + (resp. all â) boundary conditions, denotedby ”+ÎČ,G (resp. ”âÎČ,G) is the largest (resp. smallest).
Theorem 2.1 enables us to define infinite-volume measures as follows. Consider thenested sequence of boxes În = [ân,n]2. For any N > 0 and any increasing event Adepending only on spins in ÎN , the sequence (”+ÎČ,În
(A))nâ„N is decreasing4. The limit,
denoted by ”+ÎČ(A), can be defined and verified to be independent on N .
In this way, ”+ÎČ is defined for increasing events depending on a finite number ofsites. It can be further extended to a probability measure on the Ï-algebra spannedby cylindrical events (events measurable in terms of a finite number of spins). Theresulting measure, denoted by ”+ÎČ , is called the infinite-volume Ising model with +boundary conditions.
Observe that one could construct (a priori) different infinite-volume measures, forinstance with â boundary conditions (the corresponding measure is denoted by ”âÎČ). Ifinfinite-volume measures are defined from a property of compatibility with finite volumemeasures, then ”+ÎČ and ”âÎČ are extremal among infinite-volume measures of parameter
ÎČ. In particular, if ”+ÎČ = ”â
ÎČ , there exists a unique infinite volume measure.The Ising model in infinite-volume exhibits a phase transition at some critical
inverse-temperature ÎČc:
Theorem 2.2. Let ÎČc =12ln(1 +â2). The magnetization ”+ÎČ[Ï0] at the origin is
strictly positive for ÎČ > ÎČc and equal to 0 when ÎČ < ÎČc.
In other words, when ÎČ > ÎČc, there is long range memory, the phase is ordered. WhenÎČ < ÎČc, the phase is called disordered. The existence of a critical temperature separatingthe ordered from the disordered phase is a relatively easy fact [Pei36] (although atthe time it was quite unexpected). Its computation is more difficult. It was identifiedwithout proof by Kramers and Wannier [KW41a, KW41b] using the duality betweenlow and high temperature expansions of the Ising model (see the argument in the nextsection). The first rigorous derivation is due to Yang [Yan52]. He uses Onsagerâsexact formula for the (infinite-volume) partition function to compute the spontaneousmagnetization of the model. This quantity provides one criterion for localizing thecritical point. The first probabilistic computation of the critical inverse-temperature isdue to Aizenman, Barsky and FernĂĄndez [ABF87]. In Subsection 7.1, we present ashort alternative proof of Theorem 2.2, using the fermionic observable.
The critical inverse-temperature has also an interpretation in terms of infinite-volume measures (these measures are called Gibbs measures). For ÎČ < ÎČc there exists aunique Gibbs measure, while for ÎČ > ÎČc there exist several. The classification of Gibbsmeasures in the ordered phase is interesting: in dimension two, any infinite-volumemeasure is a convex combination of ”+ÎČ and ”âÎČ (see [Aiz80, Hig81] or the recent proof
[CV10]). This result is no longer true in higher dimension: non-translational-invariantGibbs measures can be constructed using 3D Dobrushin domains [Dob72].
When ÎČ > ÎČc, spin-correlations ”+ÎČ[Ï0Ïx] do not go to 0 when x goes to infinity.There is long range memory. At ÎČc, spin-correlations decay to 0 following a power law[Ons44]:
”+ÎČc[Ï0Ïx] â âŁxâŁâ1/4
when x â â. When ÎČ < ÎČc, spin-correlations decay exponentially fast in âŁxâŁ. Moreprecisely, we will show the following result first due to [MW73]:
4Indeed, for any configuration of spins in âÎn being smaller than all +, the restriction of ”+ÎČ,În+1
to În is stochastically dominated by ”+ÎČ,În
.
222 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
Theorem 2.3. For ÎČ < ÎČc, and a = eiÏ/4(x + iy) â C,
ÏÎČ(a) = limnââ
â1
nln”+ÎČ[Ï0Ï[na]] = x arcsinh(sx) + y arcsinh(sy)
where [na] is the site of L closest to na, and s solves the equationâ1 + s2x2 +
â1 + s2y2 = sinh(2ÎČ) + (sinh(2ÎČ))â1 .
The quantity ÏÎČ(z) is called the correlation length in direction z. When gettingcloser to the critical point, the correlation length goes to infinity and becomes isotropic(it does not depend on the direction, thus giving a glimpse of rotational invariance atcriticality):
Theorem 2.4 (see e.g. [Mes06]). For z â C, the correlation length satisfies thefollowing equality
limÎČÎČc
ÏÎČ(z)(ÎČc â ÎČ) = 4âŁzâŁ.(2.5)
2.2. Low and high temperature expansions of the Ising model. The lowtemperature expansion of the Ising model is a graphical representation on the dual lattice.Fix a spin configuration Ï for the Ising model on G with + boundary conditions. Thecollection of contours of a spin configuration Ï is the set of interfaces (edges of the dualgraph) separating + and â clusters. In a collection of contours, an even number of dualedges automatically emanates from each dual vertex. Reciprocally, any family of dualedges with an even number of edges emanating from each dual vertex is the collectionof contours of exactly one spin configuration (since we fix + boundary conditions).
The interesting feature of the low temperature expansion is that properties of theIsing model can be restated in terms of this graphical representation. We only give theexample of the partition function on G but other quantities can be computed similarly.Let EGâ be the set of possible collections of contours, and let âŁÏ⣠be the number of edgesof a collection of contours Ï, then
Z+ÎČ,G = eÎČ# edges in G
â âÏâEGâ
(eâ2ÎČ)âŁÏ⣠.(2.6)
The high temperature expansion of the Ising model is a graphical representation onthe primal lattice itself. It is not a geometric representation since one cannot map a spinconfiguration Ï to a subset of configurations in the graphical representation, but rathera convenient way to represent correlations between spins using statistics of contours. Itis based on the following identity:
eÎČÏxÏy = cosh(ÎČ) + ÏxÏy sinh(ÎČ) = cosh(ÎČ) [1 + tanh(ÎČ)ÏxÏy](2.7)
Proposition 2.5. Let G be a finite graph and a, b be two sites of G. At inverse-temperature ÎČ > 0,
ZfÎČ,G= 2# vertices G cosh(ÎČ)# edges in G â
ÏâEGtanh(ÎČ)âŁÏâŁ(2.8)
”fÎČ,G[ÏaÏb] = âÏâEG(a,b) tanh(ÎČ)âŁÏâŁâÏâEG tanh(ÎČ)âŁÏ⣠,(2.9)
where EG (resp. EG(a, b)) is the set of families of edges of G such that an even numberof edges emanates from each vertex (resp. except at a and b, where an odd number ofedges emanates).
The notation EG coincides with the definition EGâ in the low temperature expansionfor the dual lattice.
CONFORMAL INVARIANCE OF LATTICE MODELS 223
Proof. Let us start with the partition function (2.8). Let E be the set of edges ofG. We know
ZfÎČ,G
= âÏ
â[xy]âE
eÎČÏxÏy
= cosh(ÎČ)# edges in GâÏ
â[xy]âE
[1 + tanh(ÎČ)ÏxÏy]= cosh(ÎČ)# edges in Gâ
Ï
âÏâE
tanh(ÎČ)âŁÏ⣠âe=[xy]âÏ
ÏxÏy
= cosh(ÎČ)# edges in G âÏâE
tanh(ÎČ)âŁÏâŁâÏ
âe=[xy]âÏ
ÏxÏy
where we used (2.7) in the second equality. Notice that âÏâe=[xy]âÏ ÏxÏy equals
2# vertices G if Ï is in EG, and 0 otherwise, hence proving (2.8).Fix a, b â G. By definition,
”fÎČ,G[ÏaÏb] = âÏ ÏaÏbeâÎČH(Ï)âÏ eâÎČH(Ï) =
âÏ ÏaÏbeâÎČH(Ï)ZfÎČ,G
,(2.10)
where H(Ï) = ââiâŒj ÏiÏj . The second identity boils down to proving that the righthand terms of (2.9) and (2.10) are equal, i.e.
(2.11) âÏ
ÏaÏbeâÎČH(Ï)
= 2# vertices G cosh(ÎČ)# edges in G âÏâEG(a,b)
tanh(ÎČ)âŁÏâŁ.The first lines of the computation for the partition function are the same, and we endup with
âÏ
ÏaÏbeâÎČH(Ï)
= cosh(ÎČ)# edges in G âÏâE
tanh(ÎČ)âŁÏâŁâÏ
ÏaÏb âe=[xy]âÏ
ÏxÏy
= 2# vertices G cosh(ÎČ)# edges in G âÏâEG(a,b)
tanh(ÎČ)âŁÏâŁsince âÏ ÏaÏbâe=[xy]âÏ ÏxÏy equals 2# vertices G if Ï â EG(a, b), and 0 otherwise. â»
The set EG is the set of collections of loops on G when forgetting the way we drawloops (since some elements of EG, like a figure eight, can be decomposed into loops inseveral ways), while EG(a, b) is the set of collections of loops on G together with onecurve from a to b.
Proposition 2.6 (Kramers-Wannier duality). Let ÎČ > 0 and define ÎČâ â (0,â) suchthat tanh(ÎČâ) = eâ2ÎČ, then for every graph G,
2 # vertices Gâ cosh(ÎČâ) # edges in Gâ Z+ÎČ,G = (eÎČ)# edges in Gâ
ZfÎČâ,Gâ
.(2.12)
Proof. When writing the contour of connected components for the Ising model with+ boundary conditions, the only edges of Lâ used are those of Gâ. Indeed, edges betweenboundary sites cannot be present since boundary spins are +. Thus, the right and left-hand side terms of (2.12) both correspond to the sum on EGâ of (eâ2ÎČ)âŁÏ⣠or equivalently
of tanh(ÎČâ)âŁÏâŁ, implying the equality (see Fig. 3). â»
We are now in a position to present the argument of Kramers and Wannier. Physi-cists expect the partition function to exhibit only one singularity, localized at the criticalpoint. If ÎČâc â ÎČc, there would be at least two singularities, at ÎČc and ÎČâc , thanks to theprevious relation between partition functions at these two temperatures. Thus, ÎČc mustequal ÎČâc , which implies ÎČc =
12ln(1 +â2). Of course, the assumption that there is a
unique singularity is hard to justify.
224 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
Figure 3. The possible collections of contours for + boundary condi-tions in the low-temperature expansion do not contain edges betweenboundary sites of G. Therefore, they correspond to collections of con-tours in EGâ , which are exactly the collection of contours involved inthe high-temperature expansion of the Ising model on Gâ with freeboundary conditions.
Exercise 2.7. Extend the low and high temperature expansions to free and + bound-ary conditions respectively. Extend the high-temperature expansion to n-point spin cor-relations.
Exercise 2.8 (Peierls argument). Use the low and high temperature expansions toshow that ÎČc â (0,â), and that correlations between spins decay exponentially fast whenÎČ is small enough.
2.3. Spin-Dobrushin domain, fermionic observable and results on theIsing model. In this section we discuss the scaling limit of a single interface between+ and â at criticality. We introduce the fundamental notions of Dobrushin domains andthe so-called fermionic observable.
Let (Ω, a, b) be a simply connected domain with two marked points on the boundary.Let ΩΎ be the medial graph of ΩΎ composed of all the vertices of LÎŽ bordering a blackface associated to ΩΎ, see Fig 4. This definition is non-standard since we include medialvertices not associated to edges of ΩΎ. Let aÎŽ and bÎŽ be two vertices of âΩΎ close to aand b. We further require that bÎŽ is the southeast corner of a black face. We call thetriplet (ΩΎ , aÎŽ, bÎŽ) a spin-Dobrushin domain.
Let zÎŽ â Ω
ÎŽ . Mimicking the high-temperature expansion of the Ising model on ΩΎ,let E(aÎŽ, zÎŽ) be the set of collections of contours drawn on ΩΎ composed of loops andone interface from aÎŽ to zÎŽ, see Fig. 4. For a loop configuration Ï, Îł(Ï) denotes theunique curve from aÎŽ to zÎŽ turning always left when there is an ambiguity. With thesenotations, we can define the spin-Ising fermionic observable.
Definition 2.9. On a spin Dobrushin domain (ΩΎ , aÎŽ, bÎŽ), the spin-Ising fermionicobservable at zÎŽ â Ω
ÎŽ is defined by
FΩΎ ,aÎŽ,bÎŽ(zÎŽ) = âÏâE(aÎŽ,zÎŽ) eâ1
2iWÎł(Ï)(aÎŽ,zÎŽ)(â2 â 1)âŁÏâŁ
âÏâE(aÎŽ,bÎŽ) eâ 1
2iWÎł(Ï)(aÎŽ,bÎŽ)(â2 â 1)âŁÏ⣠,
where the winding WÎł(aÎŽ, zÎŽ) is the (signed) total rotation in radians of the curve Îł
between aÎŽ and zÎŽ.
The complex modulus of the denominator of the fermionic observable is connectedto the partition function of a conditioned critical Ising model. Indeed, fix bÎŽ â âΩ
ÎŽ .Even though E(aÎŽ, bÎŽ) is not exactly a high-temperature expansion (since there are two
CONFORMAL INVARIANCE OF LATTICE MODELS 225
bÎŽ
aÎŽ zÎŽ
Figure 4. An example of collection of contours in E(aΎ, zΎ) on thelattice Ω.
bÎŽ
aÎŽ
Figure 5. A high temperature expansion of an Ising model on theprimal lattice together with the corresponding configuration on the duallattice. The constraint that aÎŽ is connected to bÎŽ corresponds to thepartition function of the Ising model with +/â boundary conditions onthe domain.
half-edges starting from aÎŽ and bÎŽ respectively), it is in bijection with the set E(a, b).Therefore, (2.11) can be used to relate the denominator of the fermionic observableto the partition function of the Ising model on the primal graph with free boundaryconditions conditioned on the fact that a and b have the same spin. Let us mention thatthe numerator of the observable also has an interpretation in terms of disorder operatorsof the critical Ising model.
226 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
The weights of edges are critical (sinceâ2 â 1 = eâ2ÎČc). Therefore, the Kramers-
Wannier duality has an enlightening interpretation here. The high-temperature expan-sion can be thought of as the low-temperature expansion of an Ising model on the dualgraph, where the dual graph is constructed by adding one layer of dual vertices aroundâG, see Fig. 5. Now, the existence of a curve between aÎŽ and bÎŽ is equivalent to theexistence of an interface between pluses and minuses in this new Ising model. Therefore,it corresponds to a model with Dobrushin boundary conditions on the dual graph. Thisfact is not surprising since the dual boundary conditions of the free boundary conditionsconditioned on Ïa = Ïb are the Dobrushin ones.
From now on, the Ising model on a spin Dobrushin domain is the critical Ising modelon ΩâÎŽ with Dobrushin boundary conditions. The previous paragraph suggests a connec-tion between the fermionic observable and the interface in this model. In fact, Section 6will show that the fermionic observable is crucial in the proof that the unique interfaceγΎ going from aÎŽ to bÎŽ between the + component connected to the arc ââab and the âcomponent connected to ââba (preserve the convention that the interface turns left everytime there is a choice) is conformally invariant in the scaling limit. Figures 1 (centerpicture) and 2 show two interfaces in domains with Dobrushin boundary conditions.
Theorem 2.10. Let (Ω, a, b) be a simply connected domain with two marked pointson the boundary. Let γΎ be the interface of the critical Ising model with Dobrushinboundary conditions on the spin Dobrushin domain (ΩΎ , aÎŽ, bÎŽ). Then (γΎ)ÎŽ>0 convergesweakly as ÎŽ â 0 to the (chordal) Schramm-Loewner Evolution with parameter Îș = 3.
The proof of Theorem 2.10 follows the program below, see Section 6:
Prove that the family of interfaces (γΎ)Ύ>0 is tight. Prove that MzΎ
t = FΩΎâγΎ[0,t],γΎ(t),bÎŽ(zÎŽ) is a martingale for the discrete curve
γΎ. Prove that these martingales are converging when Ύ goes to 0. This provides us
with a continuous martingale (Mzt )t for any sub-sequential limit of the family(γΎ)Ύ>0.
Use the martingales (Mzt )t to identify the possible sub-sequential limits. Ac-
tually, we will prove that the (chordal) Schramm-Loewner Evolution with pa-rameter Îș = 3 is the only possible limit, thus proving the convergence.
The third step (convergence of the observable) will be crucial for the success of thisprogram. We state it as a theorem on its own. The connection with the other steps willbe explained in detail in Section 6.
Theorem 2.11 ([CS09]). Let Ω be a simply connected domain and a, b two markedpoints on its boundary, assuming that the boundary is smooth in a neighborhood of b.We have that
FΩΎ ,aÎŽ,bÎŽ(â ) âÂżĂĂĂ ÏâČ(â )
ÏâČ(b) when ÎŽ â 0(2.13)
uniformly on every compact subset of Ω, where Ï is any conformal map from Ω to theupper half-plane H, mapping a to â and b to 0.
The fermionic observable is a powerful tool to prove conformal invariance, yet itis also interesting in itself. Being defined in terms of the high-temperature expansionof the Ising model, it expresses directly quantities of the model. For instance, we willexplain in Section 6 how a more general convergence result for the observable enablesus to compute the energy density.
CONFORMAL INVARIANCE OF LATTICE MODELS 227
Theorem 2.12 ([HS10]). Let Ω be a simply connected domain and a â Ω. If eÎŽ =[xy] denotes the edge of Ω â© ÎŽZ2 closest to a, then the following equality holds:
”f
ÎČc,Ωâ©ÎŽZ2 [ÏxÏy] =â2
2âÏâČa(a)Ï
Ύ + o(Ύ),where ”f
ÎČc,ΩΎis the Ising measure at criticality and Ïa is the unique conformal map from
Ω to the disk D sending a to 0 and such that ÏâČa(a) > 0.3. Two-dimensional FK-Ising model
In this section, another graphical representation of the Ising model, called the FK-Ising model, is presented in detail. Its properties will be used to describe properties ofthe Ising model in the following sections.
3.1. FK percolation. We refer to [Gri06] for a complete study on FK percolation(invented by Fortuin and Kasteleyn [FK72]). A configuration Ï on G is a randomsubgraph of G, composed of the same sites and a subset of its edges. The edges belongingto Ï are called open, the others closed. Two sites x and y are said to be connected(denoted by x â y), if there is an open path â a path composed of open edges âconnecting them. The maximal connected components are called clusters.
Boundary conditions Ο are given by a partition of âG. Let o(Ï) (resp. c(Ï)) denotethe number of open (resp. closed) edges of Ï and k(Ï, Ο) the number of connectedcomponents of the graph obtained from Ï by identifying (or wiring) the vertices in Ο
that belong to the same class of Ο.
The FK percolation ÏΟp,q,G on a finite graph G with parameters p â [0,1], and
q â (0,â) and boundary conditions Ο is defined by
(3.1) ÏΟp,q,G(Ï) â¶= po(Ï)(1 â p)c(Ï)qk(Ï,Ο)
ZΟp,q,G
,
for any subgraph Ï of G, where ZΟp,q,G is a normalizing constant called the partitionfunction for the FK percolation. Here and in the following, we drop the dependence onΟ in k(Ï, Ο).
The FK percolations with parameter q < 1 and q â„ 1 behave very differently. Fornow, we restrict ourselves to the second case. When q â„ 1, the FK percolation is positivelycorrelated : an event is called increasing if it is preserved by addition of open edges.
Theorem 3.1. For q â„ 1 and p â [0,1], the FK percolation on G satisfies thefollowing two properties:
FKG inequality: For any boundary conditions Ο and any increasing eventsA,B,
ÏΟp,q,G(A â©B) â„ Ï
Οp,q,G(A)ÏΟp,q,G(B).(3.2)
Comparison between boundary conditions: for any Ο refinement of Ïand any increasing event A,
(3.3) ÏÏp,q,G(A) â„ Ï
Οp,q,G(A).
The previous result is very similar to Theorem 2.1. As in the Ising model case, onecan define a notion of stochastic domination. Two boundary conditions play a specialrole in the study of FK percolation: the wired boundary conditions, denoted by Ο = 1,are specified by the fact that all the vertices on the boundary are pairwise connected.The free boundary conditions, denoted by Ο = 0, are specified by the absence of wiringsbetween boundary sites. The free and wired boundary conditions are extremal amongall boundary conditions for stochastic ordering.
228 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
Infinite-volume measures can be defined as limits of measures on nested boxes. Inparticular, we set Ï1p,q for the infinite-volume measure with wired boundary conditions
and Ï0p,q for the infinite-volume measure with free boundary conditions. Like the Isingmodel, the model exhibits a phase transition in the infinite-volume limit.
Theorem 3.2. For any q â„ 1, there exists pc(q) â (0,1) such that for any infinitevolume measure Ïp,q,
if p < pc(q), there is almost surely no infinite cluster under Ïp,q, if p > pc(q), there is almost surely a unique infinite cluster under Ïp,q.
Note that q = 1 is simply bond percolation. In this case, the existence of a phasetransition is a well-known fact. The existence of a critical point in the general case q â„ 1is not much harder to prove: a coupling between two measures Ïp1,q,G and Ïp2,q,G canbe constructed in such a way that Ïp1,q,G stochastically dominates Ïp2,q,G if p1 â„ p2(this coupling is not as straightforward as in the percolation case, see e.g. [Gri06]).The determination of the critical value is a much harder task.
A natural notion of duality also exists for the FK percolation on the square lattice(and more generally on any planar graph). We present duality in the simplest case ofwired boundary conditions. Construct a model on Gâ by declaring any edge of the dualgraph to be open (resp. closed) if the corresponding edge of the primal graph is closed(resp. open) for the initial FK percolation model.
Proposition 3.3. The dual model of the FK percolation with parameters (p, q)with wired boundary conditions is the FK percolation with parameters (pâ, q) and freeboundary conditions on Gâ, where
pâ = pâ(p, q) â¶= (1 â p)q(1 â p)q + p(3.4)
Proof. Note that the state of edges between two sites of âG is not relevant whenboundary conditions are wired. Indeed, sites on the boundary are connected via bound-ary conditions anyway, so that the state of each boundary edge does not alter theconnectivity properties of the subgraph, and is independent of other edges. For thisreason, forget about edges between boundary sites and consider only inner edges (whichcorrespond to edges of Gâ): o(Ï) and c(Ï) then denote the number of open and closedinner edges.
Set eâ for the dual edge of Gâ associated to the (inner) edge e. From the definition ofthe dual configuration Ïâ of Ï, we have o(Ïâ) = aâo(Ï) where a is the number of edgesin Gâ and o(Ïâ) is the number of open dual edges. Moreover, connected components ofÏâ correspond exactly to faces of Ï, so that f(Ï) = k(Ïâ), where f(Ï) is the number offaces (counting the infinite face). Using Eulerâs formula
# edges + # connected components + 1 = #sites + # faces,
which is valid for any planar graph, we obtain, with s being the number of sites in G,
k(Ï) = s â 1 + f(Ï) â o(Ï) = s â 1 + k(Ïâ) â a + o(Ïâ).
CONFORMAL INVARIANCE OF LATTICE MODELS 229
The probability of Ïâ is equal to the probability of Ï under Ï1G,p,q, i.e.
Ï1G,p,q(Ï) = 1
Z1G,p,q
po(Ï)(1 â p)c(Ï)qk(Ï)=(1 â p)aZ1G,p,q
[p/(1 â p)]o(Ï)qk(Ï)=(1 â p)aZ1G,p,q
[p/(1 â p)]aâo(Ïâ)qsâ1âa+k(Ïâ)+o(Ïâ)=
paqsâ1âa
Z1G,p,q
[q(1 â p)/p]o(Ïâ)qk(Ïâ) = Ï0pâ,q,Gâ(Ïâ)since q(1 â p)/p = pâ/(1 â pâ), which is exactly the statement. â»
It is then natural to define the self-dual point psd = psd(q) solving the equationpâsd = psd, which gives
psd = psd(q) â¶= âq
1 +âq.
Note that, mimicking the Kramers-Wannier argument, one can give a simple heuristicjustification in favor of pc(q) = psd(q). Recently, the computation of pc(q) was performedfor every q â„ 1:
Theorem 3.4 ([BDC10]). The critical parameter pc(q) of the FK percolation onthe square lattice equals psd(q) =âq/(1 +âq) for every q â„ 1.
Exercise 3.5. Describe the dual of a FK percolation with parameters (p, q) and freeboundary conditions. What is the dual model of the FK percolation in infinite-volumewith wired boundary conditions?
Exercise 3.6 (Zhangâs argument for FK percolation, [Gri06]). Consider the FKpercolation with parameters q â„ 1 and p = psd(q). We suppose known the fact that infiniteclusters are unique, and that the probability that there is an infinite cluster is 0 or 1.
Assume that there is a.s. an infinite cluster for the measure Ï0psd,q.
1) Let Δ < 1/100. Show that there exists n > 0 such that the Ï0psd,q-probability that
the infinite cluster touches [ân,n]2 is larger than 1 â Δ. Using the FKG inequality fordecreasing events (one can check that the FKG inequality holds for decreasing events aswell), show that the Ï0psd,q-probability that the infinite cluster touches nĂ [ân,n] from
the outside of [ân,n]2 is larger than 1 â Δ1
4 .2) Using the uniqueness of the infinite cluster and the fact that the probability that
there exists an infinite cluster equals 0 or 1 (can you prove these facts?), show that a.s.there is no infinite cluster for the FK percolation with free boundary conditions at theself-dual point.
3) Is the previous result necessarily true for the FK percolation with wired boundaryconditions at the self-dual point? What can be said about pc(q)?
Exercise 3.7. Prove Eulerâs formula.
3.2. FK-Ising model and Edwards-Sokal coupling. The Ising model can becoupled to the FK percolation with cluster-weight q = 2 [ES88]. For this reason, the q = 2FK percolation model will be called the FK-Ising model. We now present this coupling,called the Edwards-Sokal coupling, along with some consequences for the Ising model.
Let G be a finite graph and let Ï be a configuration of open and closed edges on G.A spin configuration Ï can be constructed on the graph G by assigning independentlyto each cluster of Ï a + or â spin with probability 1/2 (note that all the sites of a clusterreceive the same spin).
230 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
Proposition 3.8. Let p â (0,1) and G a finite graph. If the configuration Ï isdistributed according to a FK measure with parameters (p,2) and free boundary condi-tions, then the spin configuration Ï is distributed according to an Ising measure withinverse-temperature ÎČ = â 1
2ln(1 â p) and free boundary conditions.
Proof. Consider a finite graphG, let p â (0,1). Consider a measure P on pairs (Ï,Ï),where Ï is a FK configuration with free boundary conditions and Ï is the correspondingrandom spin configuration, constructed as explained above. Then, for (Ï,Ï), we have:
P [(Ï,Ï)] = 1
Z0p,2,G
po(Ï)(1 â p)c(Ï)2k(Ï) â 2âk(Ï) = 1
Z0p,2,G
po(Ï)(1 â p)c(Ï).Now, we construct another measure P on pairs of percolation configurations and spinconfigurations as follows. Let Ï be a spin configuration distributed according to an Isingmodel with inverse-temperature ÎČ satisfying eâ2ÎČ = 1 â p and free boundary conditions.We deduce Ï from Ï by closing all edges between neighboring sites with different spins,and by independently opening with probability p edges between neighboring sites withsame spins. Then, for any (Ï, Ï),
P [(Ï, Ï)] = eâ2ÎČr(Ï)po(Ï)(1 â p)aâo(Ï)âr(Ï)ZfÎČ,p
=po(Ï)(1 â p)c(Ï)
ZfÎČ,p
where a is the number of edges of G and r(Ï) the number of edges between sites withdifferent spins.
Note that the two previous measures are in fact defined on the same set of compatiblepairs of configurations: if Ï has been obtained from Ï, then Ï can be obtained from Ï
via the second procedure described above, and the same is true in the reverse directionfor Ï and Ï. Therefore, P = P and the marginals of P are the FK percolation withparameters (p,2) and the Ising model at inverse-temperature ÎČ, which is the claim. â»
The coupling gives a randomized procedure to obtain a spin-Ising configurationfrom a FK-Ising configuration (it suffices to assign random spins). The proof of Propo-sition 3.8 provides a randomized procedure to obtain a FK-Ising configuration from aspin-Ising configuration.
If one considers wired boundary conditions for the FK percolation, the Edwards-Sokal coupling provides us with an Ising configuration with + boundary conditions (orâ, the two cases being symmetric). We do not enter into details, since the generalizationis straightforward.
An important consequence of the Edwards-Sokal coupling is the relation betweenIsing correlations and FK connectivity properties. Indeed, two sites which are connectedin the FK percolation configuration must have the same spin, while sites which are nothave independent spins. This implies:
Corollary 3.9. For p â (0,1), G a finite graph and ÎČ = â 12ln(1 â p), we obtain
”fÎČ,G[ÏxÏy] = Ï0p,2,G(xâ y),
”+ÎČ,G[Ïx] = Ï1p,2,G(xâ âG).In particular, ÎČc = â
12ln[1 â pc(2)].
Proof. We leave the proof as an exercise. â»
The uniqueness of Ising infinite-volume measures was discussed in the previous sec-tion. The same question can be asked in the case of the FK-Ising model. First, it can beproved that Ï1p,2 and Ï0p,2 are extremal among all infinite-volume measures. Therefore,
it is sufficient to prove that Ï1p,2 = Ï0p,2 to prove uniqueness. Second, the absence of
an infinite cluster for Ï1p,2 can be shown to imply the uniqueness of the infinite-volume
CONFORMAL INVARIANCE OF LATTICE MODELS 231
measure. Using the equality pc = psd, the measure is necessarily unique whenever p < psdsince Ï1p,2 has no infinite cluster. Planar duality shows that the only value of p for which
uniqueness could eventually fail is the (critical) self-dual pointâ2/(1 +â2). It turns
out that even for this value, there exists a unique infinite volume measure. Since thisfact will play a role in the proof of conformal invariance, we now sketch an elementaryproof due to W. Werner (the complete proof can be found in [Wer09]).
Proposition 3.10. There exists a unique infinite-volume FK-Ising measure withparameter pc =
â2/(1 +â2) and there is almost surely no infinite cluster under this
measure. Correspondingly, there exists a unique infinite-volume spin Ising measure atÎČc.
Proof. As described above, it is sufficient to prove that Ï0psd,2 = Ï1psd,2
. First note
that there is no infinite cluster for Ï0psd,2 thanks to Exercise 3.6. Via the Edwards-Sokalcoupling, the infinite-volume Ising measure with free boundary conditions, denoted by
”fÎČc
, can be constructed by coloring clusters of the measure Ï0psd,2. Since there is no
infinite cluster, this measure is obviously symmetric by global exchange of +/â. Inparticular, the argument of Exercise 3.6 can be applied to prove that there are neither+ nor â infinite clusters. Therefore, fixing a box, there exists a + star-connected circuitsurrounding the box with probability one (two vertices x and y are said to be star-connected if y is one of the eight closest neighbors to x).
One can then argue that the configuration inside the box stochastically dominatesthe Ising configuration for the infinite-volume measure with + boundary conditions(roughly speaking, the circuit of spin + behaves like + boundary conditions). We deduce
that ”fÎČc
restricted to the box (in fact to any box) stochastically dominates ”+ÎČc. This
implies that ”fÎČc℠”+ÎČc
. Since the other inequality is obvious, ”fÎČc
and ”+ÎČcare equal.
Via Edwards-Sokalâs coupling again, Ï0psd,2 = Ï1psd,2
and there is no infinite cluster
at criticality. Moreover, ”âÎČc= ”
fÎČc= ”+ÎČc
and there is a unique infinite-volume Ising
measure at criticality. â»
Remark 3.11. More generally, the FK percolation with integer parameter q â„ 2
can be coupled with Potts models. Many properties of Potts models are derived usingFK percolation, since we have the FKG inequality at our disposal, while there is noequivalent of the spin-Ising FKG inequality for Potts models.
3.3. Loop representation of the FK-Ising model and fermionic observable.Let (Ω, a, b) be a simply connected domain with two marked points on the boundary.Let ΩΎ be an approximation of Ω, and let âab and âba denote the counterclockwise arcsin the boundary âΩΎ joining a to b (resp. b to a). We consider a FK-Ising measurewith wired boundary conditions on âba â all the edges are pairwise connected â andfree boundary conditions on the arc âab. These boundary conditions are called the
Dobrushin boundary conditions. We denote by Ïa,bΩΎ ,p
the associated FK-Ising measurewith parameter p.
The dual boundary arc ââba is the set of sites of ΩâÎŽ adjacent to âba while the dualboundary arc ââab is the set of sites of LâÎŽâΩ
â
ÎŽ adjacent to âab, see Fig. 6. A FK-Dobrushindomain (ΩΎ , aÎŽ, bÎŽ) is given by
a medial graph ΩΎ defined as the set of medial vertices associated to edges ofΩΎ and to dual edges of ââab, medial sites aÎŽ, bÎŽ â Ω
ÎŽ between arcs âba an ââab, see Fig. 6 again,
with the additional condition that bÎŽ is the southeast corner of a black face belongingto the domain.
232 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
âba
ââba
âab
ââab
멉
ÎŽ
Ado not belongto 멉
ÎŽ
Aâ
aÎŽbÎŽ
Figure 6. A domain ΩΎ with Dobrushin boundary conditions: thevertices of the primal graph are black, the vertices of the dual graph ΩâÎŽare white, and between them lies the medial graph ΩΎ . The arcs âbaand ââab are the two outermost arcs. Moreover, arcs ââba and âab are thearcs bordering âba and ââab from the inside. The arcs âab and âba (resp.ââab and ââba) are drawn in solid lines (resp. dashed lines)
Remark 3.12. Note that the definition of ΩΎ is not the same as in Section 1.2.1since we added medial vertices associated to dual edges of ââab. We chose this definitionto make sites of the dual and the primal lattices play symmetric roles. The condition thatbÎŽ is the south corner of a black face belonging to the domain is a technical condition.
Let (ΩΎ , aΎ, bΎ) be a FK-Dobrushin domain. For any FK-Ising configuration withDobrushin boundary conditions on ΩΎ, we construct a loop configuration on ΩΎ asfollows: The interfaces between the primal clusters and the dual clusters (i.e clusters inthe dual model) form a family of loops together with a path from aΎ to bΎ. The loops aredrawn as shown in Figure 7 following the edges of the medial lattice. The orientationof the medial lattice naturally gives an orientation to the loops, so that we are workingwith a model of oriented loops on the medial lattice.
The curve from aÎŽ to bÎŽ is called the exploration path and denoted by Îł = Îł(Ï).It is the interface between the open cluster connected to âba and the dual-open clusterconnected to ââab. As in the Ising model case, one can study its scaling limit when themesh size goes to 0:
CONFORMAL INVARIANCE OF LATTICE MODELS 233
bÎŽaÎŽ
âba
ââab
Îł := Îł(Ï)
ΩΎ and ΩâÎŽ âȘ ââ
ab
Figure 7. A FK percolation configuration in the Dobrushin domain(ΩΎ, aΎ, bΎ), together with the corresponding interfaces on the mediallattice: the loops are grey, and the exploration path γ from aΎ to bΎ isblack. Note that the exploration path is the interface between the opencluster connected to the wired arc and the dual-open cluster connectedto the white faces of the free arc.
Theorem 3.13 (Conformal invariance of the FK-Ising model, [KS10, CDHKS12]).Let Ω be a simply connected domain with two marked points a, b on the boundary.Let γΎ be the interface of the critical FK-Ising with Dobrushin boundary conditions on(ΩΎ, aÎŽ, bÎŽ). Then the law of γΎ converges weakly, when ÎŽ â 0, to the chordal Schramm-Loewner Evolution with Îș = 16/3.
As in the Ising model case, the proof of this theorem also involves a discrete observ-able, which converges to a conformally invariant object. We define it now.
Definition 3.14. The edge FK fermionic observable is defined on edges of ΩΎ by
(3.5) FΩΎ,aΎ,bΎ,p(e) = EaΎ,bΎΩΎ,p
[e 1
2â iWÎł(e,bÎŽ)1eâÎł],
where WÎł(e, bÎŽ) denotes the winding between the center of e and bÎŽ.The vertex FK fermionic observable is defined on vertices of ΩΎ â âΩ
ÎŽ by
FΩΎ,aΎ,bΎ,p(v) = 1
2âeâŒv
FΩΎ,aΎ,bΎ,p(e)(3.6)
where the sum is over the four medial edges having v as an endpoint.
234 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
When we consider the observable at criticality (which will be almost always thecase), we drop the dependence on p in the notation. More generally, if (Ω, a, b) is fixed,we simply denote the observable on (ΩΎ , aΎ, bΎ, psd) by FΎ.
The quantity FÎŽ(e) is a complexified version of the probability that e belongs tothe exploration path. The complex weight makes the link between FÎŽ and probabilisticproperties less explicit. Nevertheless, the vertex fermionic observable FÎŽ converges whenÎŽ goes to 0:
Theorem 3.15. [Smi10a] Let (Ω, a, b) be a simply connected domain with twomarked points on the boundary. Let FΎ be the vertex fermionic observable in (ΩΎ , aΎ, bΎ).Then, we have
1â2ÎŽFÎŽ(â ) â âÏâČ(â ) when ÎŽ â 0(3.7)
uniformly on any compact subset of Ω, where Ï is any conformal map from Ω to thestrip R Ă (0,1) mapping a to ââ and b to â.
As in the case of the spin Ising model, this statement is the heart of the proof ofconformal invariance. Yet, the observable itself can be helpful for the understanding ofother properties of the FK-Ising model. For instance, it enables us to prove a statementequivalent to the celebrated Russo-Seymour-Welsh Theorem for percolation. This resultwill be central for the proof of compactness of exploration paths (an important step inthe proof of Theorems 2.10 and 3.13).
Theorem 3.16 (RSW-type crossing bounds, [DCHN10]). There exists a constantc > 0 such that for any rectangle R of size 4n Ă n, one has
Ï0psd,2,R(there exists an open path from left to right) â„ c.(3.8)
Before ending this section, we present a simple yet crucial result: we show that itis possible to compute rather explicitly the distribution of the loop representation. Inparticular, at criticality, the weight of a loop configuration depends only on the numberof loops.
Proposition 3.17. Let p â (0,1) and let (ΩΎ , aÎŽ, bÎŽ) be a FK Dobrushin domain,then for any configuration Ï,
ÏaÎŽ,bΎΩΎ,p(Ï) = 1
Zxo(Ï)
â2â(Ï)
(3.9)
where x = p/[â2(1âp)], â(Ï) is the number of loops in the loop configuration associatedto Ï, o(Ï) is the number of open edges, and Z is the normalization constant.
Proof. Recall that
ÏaÎŽ,bΎΩΎ,p(Ï) = 1
Z[p/(1 â p)]o(Ï)2k(Ï).
Using arguments similar to Proposition 3.3, the dual of ÏaÎŽ,bΎΩΎ,p
can be proved to be ÏbÎŽ,aΎΩâ
ÎŽ,pâ
(in this sense, Dobrushin boundary conditions are self-dual). With Ïâ being the dual
CONFORMAL INVARIANCE OF LATTICE MODELS 235
configuration of Ï, we find
ÏaÎŽ,bΎΩΎ,p(Ï) = â
ÏaÎŽ,bΎΩΎ,p(Ï) ÏbÎŽ,aÎŽ
ΩâÎŽ,pâ(Ïâ)
=1âZZâ
âp/(1 â p) o(Ï)â2 k(Ï)â
pâ/(1 â pâ) o(Ïâ)â2 k(Ïâ)
=1âZZâ
ÂżĂĂĂp(1 â pâ)(1 â p)pâo(Ï)â
pâ/(1 â pâ) o(Ïâ)+o(Ï)â2 k(Ï)+k(Ïâ)
=
â2âpâ/(1 â pâ) o(Ï)+o(Ïâ)â
ZZâxo(Ï)
â2k(Ï)+k(Ïâ)â1
where the definition of pâ was used to prove thatp(1âpâ)(1âp)pâ = x
2. Note that â(Ï) =k(Ï) + k(Ïâ) â 1 and
Z =
âZZââ
2âpâ/(1 â pâ) o(Ï)+o(Ïâ)
does not depend on the configuration (the sum o(Ï) + o(Ïâ) being equal to the totalnumber of edges). Altogether, this implies the claim. â»
4. Discrete complex analysis on graphs
Complex analysis is the study of harmonic and holomorphic functions in complexdomains. In this section, we shall discuss how to discretize harmonic and holomorphicfunctions, and what are the properties of these discretizations.
There are many ways to introduce discrete structures on graphs which can be de-veloped in parallel to the usual complex analysis. We need to consider scaling limits (asthe mesh of the lattice tends to zero), so we want to deal with discrete structures whichconverge to the continuous complex analysis as finer and finer graphs are taken.
4.1. Preharmonic functions.4.1.1. Definition and connection with random walks. Introduce the (non-normalized)
discretization of the Laplacian operator â â¶= 14(â2xx + â2yy) in the case of the square
lattice LÎŽ. For u â LÎŽ and f ⶠLÎŽ â C, define
âÎŽf(u) = 1
4âvâŒu
(f(v) â f(u)).The definition extends to rescaled square lattices in a straightforward way (for instanceto L
ÎŽ ).
Definition 4.1. A function h ⶠΩΎ â C is preharmonic (resp. pre-superharmonic,pre-subharmonic) if âÎŽh(x) = 0 (resp. †0, â„ 0) for every x â ΩΎ.
One fundamental tool in the study of preharmonic functions is the classical relationbetween preharmonic functions and simple random walks:
Let (Xn) be a simple random walk killed at the first time it exits ΩΎ; then h ispreharmonic on ΩΎ if and only if (h(Xn)) is a martingale.
Using this fact, one can prove that harmonic functions are determined by their valueon âΩΎ, that they satisfy Harnackâs principle, etc. We refer to [Law91] for a deeperstudy on preharmonic functions and their link to random walks. Also note that the setof preharmonic functions is a complex vector space. As in the continuum, it is easy tosee that preharmonic functions satisfy the maximum and minimum principles.
236 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
4.1.2. Derivative estimates and compactness criteria. For general functions, a con-trol on the gradient provides regularity estimates on the function itself. It is a well-knownfact that harmonic functions satisfy the reverse property: controlling the function al-lows us to control the gradient. The following lemma shows that the same is true forpreharmonic functions.
Proposition 4.2. There exists C > 0 such that, for any preharmonic functionh ⶠΩΎ â C and any two neighboring sites x, y â ΩΎ,
âŁh(x) â h(y)⣠†CÎŽsupzâΩΎ
âŁh(z)âŁd(x,Ωc) .(4.1)
Proof. Let x, y â ΩΎ. The preharmonicity of h translates to the fact that h(Xn) isa martingale (where Xn is a simple random walk killed at the first time it exits ΩΎ).Therefore, for x, y two neighboring sites of ΩΎ, we have
h(x) â h(y) = E[h(XÏ ) â h(YÏ âČ)](4.2)
where under E, X and Y are two simple random walks starting respectively at x andy, and Ï , Ï âČ are any stopping times. Let 2r = d(x,Ωc) > 0, so that U = x + [âr, r]2 isincluded in ΩΎ. Fix Ï and Ï âČ to be the hitting times of âUÎŽ and consider the followingcoupling of X and Y (one has complete freedom in the choice of the joint law in (4.2)):(Xn) is a simple random walk and Yn is constructed as follows,
if X1 = y, then Yn =Xn+1 for n â„ 0, if X1 â y, then Yn = Ï(Xn+1), where Ï is the orthogonal symmetry with respect
to the perpendicular bisector â of [X1, y], whenever Xn+1 does not reach â. Assoon as it does, set Yn =Xn+1.
It is easy to check that Y is also a simple random walk. Moreover, we have
âŁh(x) â h(y)⣠†E[âŁh(XÏ) â h(YÏ âČ)âŁ1XÏâ YÏâČ] †2( sup
zââUÎŽ
âŁh(z)âŁ) P(XÏ â YÏ âČ)Using the definition of the coupling, the probability on the right is known: it is equalto the probability that X does not touch â before exiting the ball and is smaller thanCâČ
rÎŽ (with CâČ a universal constant), since UÎŽ is of radius r/ÎŽ for the graph distance. We
deduce that
âŁh(x) â h(y)⣠†2( supzââUÎŽ
âŁh(z)âŁ) CâČrÎŽ †2(sup
zâΩΎ
âŁh(z)âŁ) CâČrÎŽ
â»
Recall that functions on ΩΎ are implicitly extended to Ω.
Proposition 4.3. A family (hΎ)Ύ>0 of preharmonic functions on the graphs ΩΎ isprecompact for the uniform topology on compact subsets of Ω if one of the followingproperties holds:
(1) (hΎ)Ύ>0 is uniformly bounded on any compact subset of Ω,or
(2) for any compact subset K of Ω, there exists M = M(K) > 0 such that for anyΎ > 0,
ÎŽ2 âxâKÎŽ
âŁhÎŽ(x)âŁ2 â€M.
CONFORMAL INVARIANCE OF LATTICE MODELS 237
Proof. Let us prove that the proposition holds under the first hypothesis and thenthat the second hypothesis implies the first one.
We are faced with a family of continuous maps hÎŽ ⶠΩ â C and we aim to applythe ArzelĂ -Ascoli theorem. It is sufficient to prove that the functions hÎŽ are uniformlyLipschitz on any compact subset since they are uniformly bounded on any compactsubset of Ω. LetK be a compact subset of Ω. Proposition 4.2 shows that âŁhÎŽ(x)âhÎŽ(y)⣠â€CKÎŽ for any two neighbors x, y âKÎŽ, where
CK = CsupÎŽ>0 supxâΩâ¶d(x,K)â€r/2 âŁhÎŽ(x)âŁ
d(K,Ωc) ,
implying that âŁhÎŽ(x) â hÎŽ(y)⣠†2CK âŁx â y⣠for any x, y âKÎŽ (not necessarily neighbors).The ArzelĂĄ-Ascoli theorem concludes the proof.
Now assume that the second hypothesis holds, and let us prove that (hÎŽ)ÎŽ>0 isbounded on any compact subset of Ω. Take K â Ω compact, let 2r = d(K,Ωc) > 0 andconsider x âKÎŽ. Using the second hypothesis, there exists k â¶= k(x) such that r
2Ύ†k †r
ÎŽ
and
ÎŽ âyââUkÎŽ
âŁhÎŽ(y)âŁ2 †2M/r,(4.3)
where UkÎŽ = x + [âÎŽk, ÎŽk]2 is the box of size k (for the graph distance) around x andM =M(y + [âr, r]2). Exercise 4.4 implies
hÎŽ(x) = âyââUkÎŽ
hÎŽ(y)HUkÎŽ(x, y)(4.4)
for every x â UÎŽk. Using the Cauchy-Schwarz inequality, we find
hÎŽ(x)2 =ââ âyââUkÎŽ
hÎŽ(y)HUkÎŽ(x, y)ââ
2
â€ââÎŽ â âyââUkÎŽ
âŁhÎŽ(y)âŁ2ââ ââ1ÎŽ â âyââUkÎŽ
HUkÎŽ(x, y)2ââ †2M/r â C
where C is a uniform constant. The last inequality used Exercise 4.5 to affirm thatHUkÎŽ
(x, y) †CÎŽ for some C = C(r) > 0. â»
Exercise 4.4. The discrete harmonic measure HΩΎ(â , y) of y â âΩΎ is the unique
harmonic function on ΩΎ â âΩΎ vanishing on the boundary âΩΎ, except at y, where itequals 1. Equivalently, HΩΎ
(x, y) is the probability that a simple random walk startingfrom x exits ΩΎ â âΩΎ through y. Show that for any harmonic function h ⶠΩΎ â C,
h = âyââΩΎ
h(y)HΩΎ(â , y).
Exercise 4.5. Prove that there exists C > 0 such that HQΎ(0, y) †CΎ for every
ÎŽ > 0 and y â âQÎŽ, where Q = [â1,1]2.4.1.3. Discrete Dirichlet problem and convergence in the scaling limit. Preharmonic
functions on square lattices of smaller and smaller mesh size were studied in a number ofpapers in the early twentieth century (see e.g. [PW23, Bou26, Lus26]), culminatingin the seminal work of Courant, Friedrichs and Lewy. It was shown in [CFL28] thatsolutions to the Dirichlet problem for a discretization of an elliptic operator converge tothe solution of the analogous continuous problem as the mesh of the lattice tends to zero.A first interesting fact is that the limit of preharmonic functions is indeed harmonic.
Proposition 4.6. Any limit of a sequence of preharmonic functions on ΩΎ converg-ing uniformly on any compact subset of Ω is harmonic in Ω.
238 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
Proof. Let (hΎ) be a sequence of preharmonic functions on ΩΎ converging to h.Via Propositions 4.2 and 4.3, ( 1
ÎŽ[hÎŽ(â + ÎŽ) â hÎŽ])ÎŽ>0 is precompact. Since âxh is the
only possible sub-sequential limit of the sequence, ( 1â2ÎŽ[hÎŽ(â + ÎŽ) â hÎŽ])ÎŽ>0 converges
(indeed its discrete primitive converges to h). Similarly, one can prove convergence ofdiscrete derivatives of any order. In particular, 0 = 1
2ÎŽ2âÎŽhÎŽ converges to 1
4[âxxh+âyyh].
Therefore, h is harmonic. â»
In particular, preharmonic functions with a given boundary value problem convergein the scaling limit to a harmonic function with the same boundary value problem in arather strong sense, including convergence of all partial derivatives. The finest result ofconvergence of discrete Dirichlet problems to the continuous ones will not be necessaryin our setting and we state the minimal required result:
Theorem 4.7. Let Ω be a simply connected domain with two marked points a andb on the boundary, and f a bounded continuous function on the boundary of Ω. LetfÎŽ ⶠâΩΎ â C be a sequence of uniformly bounded functions converging uniformly awayfrom a and b to f . Let hÎŽ be the unique preharmonic map on ΩΎ such that (hÎŽ)âŁâΩΎ
= fÎŽ.Then
hÎŽ Ăâ h when ÎŽ â 0
uniformly on compact subsets of Ω, where h is the unique harmonic function on Ω,continuous on Ω, satisfying hâŁâΩ = f .
Proof. Since (fÎŽ)ÎŽ>0 is uniformly bounded by some constant M , the minimum andmaximum principles imply that (hÎŽ)ÎŽ>0 is bounded by M . Therefore, the family (hÎŽ) is
precompact (Proposition 4.3). Let h be a sub-sequential limit. Necessarily, h is harmonic
inside the domain (Proposition 4.6) and bounded. To prove that h = h, it suffices to
show that h can be continuously extended to the boundary by f .Let x â âΩ â a, b and Δ > 0. There exists R > 0 such that for ÎŽ small enough,
âŁfÎŽ(xâČ) â fÎŽ(x)⣠< Δ for every xâČ â âΩ â©Q(x,R),where Q(x,R) = x + [âR,R]2. For r < R and y â Q(x, r), we have
âŁhÎŽ(y) â fÎŽ(x)⣠= Ey[fÎŽ(XÏ) â fÎŽ(x)]for X a random walk starting at y, and Ï its hitting time of the boundary. Decomposingbetween walks exiting the domain inside Q(x,R) and others, we find
âŁhÎŽ(y) â fÎŽ(x)⣠†Δ + 2MPy[XÏ â Q(x,R)]Exercise 4.8 guarantees that Py[XÏ â Q(x,R)] †(r/R)α for some independent constant
α > 0. Taking r = R(Δ/2M)1/α and letting ÎŽ go to 0, we obtain âŁh(y) â f(x)⣠†2Δ forevery y â Q(x, r). â»
Exercise 4.8. Show that there exists α > 0 such that for any 1â« r > ÎŽ > 0 and anycurve Îł inside D â¶= z ⶠâŁz⣠< 1 from C = z ⶠâŁz⣠= 1 to z ⶠâŁz⣠= r, the probability for arandom walk on DÎŽ starting at 0 to exit (DâÎł)ÎŽ through C is smaller than rα. To provethis, one can show that in any annulus z ⶠx †âŁz⣠†2x, the random walk trajectory hasa uniformly positive probability to close a loop around the origin.
4.1.4. Discrete Green functions. This paragraph concludes the section by mention-ing the important example of discrete Green functions. For y â ΩΎ â âΩΎ, let GΩΎ
(â , y)be the discrete Green function in the domain ΩΎ with singularity at y, i.e. the uniquefunction on ΩΎ such that
its Laplacian on ΩΎ â âΩΎ equals 0 except at y, where it equals 1, GΩΎ
(â , y) vanishes on the boundary âΩΎ.
CONFORMAL INVARIANCE OF LATTICE MODELS 239
The quantity âGΩΎ(x, y) is the number of visits at x of a random walk started at y and
stopped at the first time it reaches the boundary. Equivalently, it is also the number ofvisits at y of a random walk started at x stopped at the first time it reaches the boundary.Green functions are very convenient, in particular because of the Riesz representationformula for (not necessarily harmonic) functions:
Proposition 4.9 (Riesz representation formula). Let f ⶠΩΎ â C be a functionvanishing on âΩΎ. We have
f = âyâΩΎ
âÎŽf(y)GΩΎ(â , y).
Proof. Note that f ââyâΩΎâÎŽf(y)GΩΎ
(â , y) is harmonic and vanishes on the bound-ary. Hence, it equals 0 everywhere. â»
Finally, a regularity estimate on discrete Green functions will be needed. Thisproposition is slightly technical. In the following, aQÎŽ = [âa, a]2 â© LÎŽ and âxf(x) =(f(x + ÎŽ) â f(x), f(x + iÎŽ) â f(x)).
Proposition 4.10. There exists C > 0 such that for any ÎŽ > 0 and y â 9QÎŽ,
âxâQÎŽ
âŁâxG9QÎŽ(x, y)⣠†CÎŽ â
xâQÎŽ
G9QÎŽ(x, y).
Proof. In the proof, C1,...,C6 denote universal constants. First assume y â 9QÎŽâ3QÎŽ.Using random walks, one can easily show that there exists C1 > 0 such that
1
C1
G9QΎ(x, y) †G9QΎ
(xâČ, y) †C1G9QÎŽ(x, y)
for every x,xâČ â 2QÎŽ (this is a special application of Harnackâs principle). Using Propo-sition 4.2, we deduce
âxâQÎŽ
âŁâxG9QÎŽ(x, y)⣠†â
xâQÎŽ
C2ÎŽ maxxâ2QÎŽ
G9QÎŽ(x, y) †C1C2ÎŽ â
xâQÎŽ
G9QÎŽ(x, y)
which is the claim for y â 9QÎŽ â 3QÎŽ.Assume now that y â 3QÎŽ. Using the fact that G9QÎŽ
(x, y) is the number of visits ofx for a random walk starting at y (and stopped on the boundary), we find
âxâQÎŽ
G9QÎŽ(x, y) â„ C3/ÎŽ2.
Therefore, it suffices to proveâxâQÎŽâŁâG9QÎŽ
(x, y)⣠†C4/Ύ. Let GLΎbe the Green function
in the whole plane, i.e. the function with Laplacian equal to ÎŽx,y, normalized so thatGLÎŽ(y, y) = 0, and with sublinear growth. This function has been widely studied, it was
proved in [MW40] that
GLÎŽ(x, y) = 1
Ïln(âŁx â yâŁ
ÎŽ) +C5 + o( ÎŽâŁx â y⣠) .
Now, GLÎŽ(â , y)âG9QÎŽ
(â , y)â 1Ïln (1
ÎŽ) is harmonic and has bounded boundary conditions
on â9QÎŽ. Therefore, Proposition 4.2 implies
âxâQÎŽ
âŁâx(GLÎŽ(x, y) âG9QÎŽ
(x, y))⣠†C6ÎŽ â 1/ÎŽ2 = C6/ÎŽ.Moreover, the asymptotic of GLÎŽ
(â , y) leads to
âxâQÎŽ
âŁâxGLÎŽ(x, y)⣠†C7/ÎŽ.
Summing the two inequalities, the result follows readily. â»
240 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
4.2. Preholomorphic functions.4.2.1. Historical introduction. Preholomorphic functions appeared implicitly in Kirch-
hoffâs work [Kir47], in which a graph is modeled as an electric network. Assume everyedge of the graph is a unit resistor and for u ⌠v, let F (uv) be the current from u to v.The first and the second Kirchhoffâs laws of electricity can be restated:
the sum of currents flowing from a vertex is zero:
âvâŒu
F (uv) = 0,(4.5)
the sum of the currents around any oriented closed contour Îł is zero:
â[uv]âÎł
F (uv) = 0.(4.6)
Different resistances amount to putting weights into (4.5) and (4.6). The secondlaw is equivalent to saying that F is given by the gradient of a potential function H ,and the first equivalent to H being preharmonic.
Besides the original work of Kirchhoff, the first notable application of preholomor-phic functions is perhaps the famous article [BSST40] of Brooks, Smith, Stone andTutte, where preholomorphic functions were used to construct tilings of rectangles bysquares.
Preholomorphic functions distinctively appeared for the first time in the papers[Isa41, Isa52] of Isaacs, where he proposed two definitions (and called such functionsmono-diffric). Both definitions ask for a discrete version of the Cauchy-Riemann equa-tions âiαF = iâαF or equivalently that the z-derivative is 0. In the first definition, theequation that the function must satisfy is
i [f (E) â f (S)] = f (W ) â f (S)while in the second, it is
i [f (E) â f (W )] = f (N) â f (S) ,where N , E, S and W are the four corners of a face. A few papers of his and othermathematicians followed, studying the first definition, which is asymmetric on the squarelattice. The second (symmetric) definition was reintroduced by Ferrand, who also dis-cussed the passage to the scaling limit and gave new proofs of Riemann uniformizationand the Courant-Friedrichs-Lewy theorems [Fer44, LF55]. This was followed by ex-tensive studies of Duffin and others, starting with [Duf56].
4.3. Isaacsâs definition of preholomorphic functions. We will be workingwith Isaacsâs second definition (although the theories based on both definitions arealmost the same). The definition involves the following discretization of the â = âx + iâyoperator. For a complex valued function f on LÎŽ (or on a finite subgraph of it), andx â LâÎŽ , define
âÎŽf(x) = 1
2[f (E) â f (W )] + i
2[f (N) â f (S)]
where N , E, S and W denote the four vertices adjacent to the dual vertex x indexed inthe obvious way.
Remark 4.11. When defining derivation, one uses duality between a graph and itsdual. Quantities related to the derivative of a function on G are defined on the dualgraph Gâ. Similarly, notions related to the second derivative are defined on the graph Gagain, whereas a primitive would be defined on Gâ.
Definition 4.12. A function f ⶠΩΎ â C is called preholomorphic if âÎŽf(x) = 0 forevery x â ΩâÎŽ . For x â ΩâÎŽ , âÎŽf(x) = 0 is called the discrete Cauchy-Riemann equation atx.
CONFORMAL INVARIANCE OF LATTICE MODELS 241
eâiÏ/4
1
eiÏ/4
i
v
NW NE
SESW
Figure 8. Lines â(e) for medial edges around a white face.
The theory of preholomorphic functions starts much like the usual complex analysis.Sums of preholomorphic functions are also preholomorphic, discrete contour integralsvanish, primitive (in a simply-connected domain) and derivative are well-defined and arepreholomorphic functions on the dual square lattice, etc. In particular, the (discrete)gradient of a preharmonic function is preholomorphic (this property has been proposedas a suitable generalization in higher dimensions).
Exercise 4.13. Prove that the restriction of a continuous holomorphic function toLÎŽ satisfies discrete Cauchy-Riemann equations up to O(ÎŽ3).
Exercise 4.14. Prove that any preholomorphic function is preharmonic for aslightly modified Laplacian (the average over edges at distance
â2ÎŽ minus the value at
the point). Prove that the (discrete) gradient of a preharmonic function is preholomor-phic (this property has been proposed as a suitable generalization in higher dimensions).Prove that the limit of preholomorphic functions is holomorphic.
Exercise 4.15. Prove that the integral of a preholomorphic function along a dis-crete contour vanishes. Prove that the primitive and the differential of preholomorphicfunctions are preholomorphic.
Exercise 4.16. Prove that 1â2ÎŽâÎŽ and 1
2ÎŽ2âÎŽ converge (when ÎŽ â 0) to â, â and â
in the sense of distributions.
4.4. s-holomorphic functions. As explained in the previous sections, the theoryof preholomorphic functions starts like the continuum theory. Unfortunately, problemsarrive quickly. For instance, the square of a preholomorphic function is no longer pre-holomorphic in general. This makes the theory of preholomorphic functions significantlyharder than the usual complex analysis, since one cannot transpose proofs from contin-uum to discrete in a straightforward way. In order to partially overcome this difficulty,we introduce s-holomorphic functions (for spin-holomorphic), a notion that will be cen-tral in the study of the spin and FK fermionic observables.
4.4.1. Definition of s-holomorphic functions. To any edge of the medial lattice e,we associate a line â(e) passing through the origin and
âe (the choice of the square root
is not important, and recall that e being oriented, it can be thought of as a complexnumber). The different lines associated with medial edges on L
ÎŽ are R, eiÏ/4R, iR and
e3iÏ/4R, see Fig. 8.
Definition 4.17. A function f ⶠΩΎ â C is s-holomorphic if for any edge e of ΩΎ ,we have
Pâ(e)[f(x)] = Pâ(e)[f(y)]where x, y are the endpoints of e and Pâ is the orthogonal projection on â.
242 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
The definition of s-holomorphicity is not rotationally invariant. Nevertheless, f iss-holomorphic if and only if eiÏ/4f(iâ ) (resp. if(ââ )) is s-holomorphic.
Proposition 4.18. Any s-holomorphic function f ⶠΩΎ â C is preholomorphic onΩΎ .
Proof. Let f ⶠΩΎ â C be a s-holomorphic function. Let v be a vertex of LÎŽ âȘ Lâ
ÎŽ
(this is the vertex set of the dual of the medial lattice). Assume that v â ΩâÎŽ , the othercase is similar. We aim to show that âÎŽf(v) = 0. Let NW , NE, SE and SW be thefour vertices around v as illustrated in Fig. 8. Next, let us write relations provided bythe s-holomorphicity, for instance
P R[f(NW )] = P R[f(NE)].Expressed in terms of f and its complex conjugate f only, we obtain
f(NW ) + f(NW ) = f(NE) + f(NE).Doing the same with the other edges, we find
f(NE) + if(NE) = f(SE) + if(SE)f(SE) â f(SE) = f(SW ) â f(SW )f(SW ) â if(SW ) = f(NW ) â if(NW )
Multiplying the second identity by âi, the third by â1, the fourth by i, and then summingthe four identities, we obtain
0 = (1 â i) [f(NW ) â f(SE)+ if(SW )â if(NE)] = 2(1 â i)âÎŽf(v)which is exactly the discrete Cauchy-Riemann equation in the medial lattice. â»
4.4.2. Discrete primitive of F 2. One might wonder why s-holomorphicity is an in-teresting concept, since it is more restrictive than preholomorphicity. The answercomes from the fact that a relevant discretization of 1
2Im (â« z f2) can be defined for
s-holomorphic functions f .
Theorem 4.19. Let f ⶠΩΎ â C be an s-holomorphic function on the discrete simplyconnected domain ΩΎ , and b0 â ΩΎ. Then, there exists a unique function H ⶠΩΎâȘΩ
â
ÎŽ â C
such that
H(b0) = 1 and
H(b) âH(w) = ÎŽ âŁPâ(e)[f(x)]âŁ2 (= ÎŽ âŁPâ(e)[f(y)]âŁ2 )for every edge e = [xy] of ΩΎ bordered by a black face b â ΩΎ and a white face w â ΩâÎŽ .
An elementary computation shows that for two neighboring sites b1, b2 â ΩΎ, with vbeing the medial vertex at the center of [b1b2],
H(b1) âH(b2) = 1
2Im [f(v)2 â (b1 â b2)] ,
the same relation holding for sites of ΩâÎŽ . This legitimizes the fact that H is a discrete
analogue of 12Im (â« z f2).
Proof. The uniqueness of H is straightforward since ΩΎ is simply connected. Toobtain the existence, construct the value at some point by summing increments alongan arbitrary path from b0 to this point. The only thing to check is that the valueobtained does not depend on the path chosen to define it. Equivalently, we must checkthe second Kirchhoffâs law. Since the domain is simply connected, it is sufficient to checkit for elementary square contours around each medial vertex v (these are the simplestclosed contours). Therefore, we need to prove that
(4.7) âŁPâ(n)[f(v)]âŁ2 â âŁPâ(e)[f(v)]âŁ2 + âŁPâ(s)[f(v)]âŁ2 â âŁPâ(w)[f(v)]âŁ2 = 0,
CONFORMAL INVARIANCE OF LATTICE MODELS 243
B
Figure 9. Arrows corresponding to contributions to 2âH. Note thatarrows from black to white contribute negatively, those from white toblack positively.
where n, e, s and w are the four medial edges with endpoint v, indexed in the obviousway. Note that â(n) and â(s) (resp. â(e) and â(w)) are orthogonal. Hence, (4.7) followsfrom
(4.8) âŁPâ(n)[f(v)]âŁ2 + âŁPâ(s)[f(v)]âŁ2 = âŁf(v)âŁ2 = âŁPâ(e)[f(v)]âŁ2 + âŁPâ(w)[f(v)]âŁ2.â»
Even if the primitive of f is preholomorphic and thus preharmonic, this is notthe case for H in general5. Nonetheless, H satisfies subharmonic and superharmonicproperties. Denote by H and H the restrictions of H ⶠΩΎ âȘΩ
â
ÎŽ â C to ΩΎ (black faces)and ΩâÎŽ (white faces).
Proposition 4.20. If f ⶠΩΎ â C is s-holomorphic, then H and H are respectivelysubharmonic and superharmonic.
Proof. Let B be a vertex of ΩΎ â âΩΎ. We aim to show that the sum of incrementsof H between B and its four neighbors is positive. In other words, we need to provethat the sum of increments along the sixteen arrows drawn in Fig. 9 is positive. Let a,b, c and d be the four values of
âÎŽPâ(e)[f(y)] for every vertex y â ΩΎ around B and any
edge e = [yz] bordering B (there are only four different values thanks to the definitionof s-holomorphicity). An easy computation shows that the eight interior increments arethus âa2, âb2, âc2, âd2 (each appearing twice). Using the s-holomorphicity of f onvertices of ΩΎ around B, we can compute the eight exterior increments in terms of a, b,
c and d: we obtain (aâ2âb)2, (bâ2âa)2, (bâ2âc)2, (câ2âb)2, (câ2âd)2, (dâ2âc)2,(dâ2 + a)2, (aâ2 + d)2. Hence, the sum S of increments equals
S = 4(a2 + b2 + c2 + d2) â 4â2(ab + bc + cd â da)(4.9)
= 4âŁeâiÏ/4a â b + ei3Ï/4c â idâŁ2 â„ 0.(4.10)
The proof for H follows along the same lines. â»
Remark 4.21. A subharmonic function in a domain is smaller than the harmonicfunction with the same boundary conditions. Therefore, H is smaller than the harmonicfunction solving the same boundary value problem while H is bigger than the harmonicfunction solving the same boundary value problem. Moreover, H(b) is larger thanH(w) for two neighboring faces. Hence, if H and H are close to each other on theboundary, then they are sandwiched between two harmonic functions with roughly the
5H is roughly (the imaginary part of) the primitive of the square of f .
244 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
Figure 10. The black graph is the isoradial graph. Grey vertices arethe vertices on the dual graph. There exists a radius r > 0 such that allfaces can be put into an incircle of radius r. Dual vertices have beendrawn in such a way that they are the centers of these circles.
same boundary conditions. In this case, they are almost harmonic. This fact will becentral in the proof of conformal invariance.
4.5. Isoradial graphs and circle packings. Duffin [Duf68] extended the defini-tion of preholomorphic functions to isoradial graphs. Isoradial graphs are planar graphsthat can be embedded in such a way that there exists r > 0 so that each face has acircumcircle of same radius r > 0, see Fig. 10. When the embedding satisfies this prop-erty, it is said to be an isoradial embedding. We would like to point out that isoradialgraphs form a rather large family of graphs. While not every topological quadrangula-tion (graph all of whose faces are quadrangles) admits a isoradial embedding, Kenyonand Schlenker [KS05] gave a simple necessary and sufficient topological condition forits existence. It seems that the first appearance of a related family of graphs in theprobabilistic context was in the work of Baxter [Bax89], where the eight-vertex modeland the Ising model were considered on Z-invariant graphs, arising from planar line ar-rangements. These graphs are topologically the same as the isoradial ones, and thoughthey are embedded differently into the plane, by [KS05] they always admit isoradialembeddings. In [Bax89], Baxter was not considering scaling limits, and so the actualchoice of embedding was immaterial for his results. However, weights in his modelswould suggest an isoradial embedding, and the Ising model was so considered by Mercat[Mer01], Boutilier and de TiliĂšre [BdT11, BdT10], Chelkak and Smirnov [CS08] (seethe last section for more details). Additionally, the dimer and the uniform spanningtree models on such graphs also have nice properties, see e.g. [Ken02]. Today, isoradialgraphs seem to be the largest family of graphs for which certain lattice models, includingthe Ising model, have nice integrability properties (for instance, the star-triangle relationworks nicely). A second reason to study isoradial graphs is that it is perhaps the largestfamily of graphs for which the Cauchy-Riemann operator admits a nice discretization.In particular, restrictions of holomorphic functions to such graphs are preholomorphicto higher orders. The fact that isoradial graphs are natural graphs both for discreteanalysis and statistical physics sheds yet another light on the connection between thetwo domains.
CONFORMAL INVARIANCE OF LATTICE MODELS 245
In [Thu86], Thurston proposed circle packings as another discretization of complexanalysis. Some beautiful applications were found, including yet another proof of the Rie-mann uniformization theorem by Rodin and Sullivan [RS87]. More interestingly, circlepackings were used by He and Schramm [HS93] in the best result so far on the Koebeuniformization conjecture, stating that any domain can be conformally uniformized toa domain bounded by circles and points. In particular, they established the conjecturefor domains with countably many boundary components. More about circle packingscan be learned from Stephensonâs book [Ste05]. Note that unlike the discretizationsdiscussed above, the circle packings lead to non-linear versions of the Cauchy-Riemannequations, see e.g. the discussion in [BMS05].
5. Convergence of fermionic observables
In this section, we prove the convergence of fermionic observables at criticality (The-orems 2.11 and 3.15). We start with the easier case of the FK-Ising model. We presentthe complete proof of the convergence, the main tool being the discrete complex analysisthat we developed in the previous section. We also sketch the proof of the convergencefor the spin Ising model.
5.1. Convergence of the FK fermionic observable. In this section, fix a sim-ply connected domain (Ω, a, b) with two points on the boundary. For Ύ > 0, always con-sider a discrete FK Dobrushin domain (ΩΎ , aΎ, bΎ) and the critical FK-Ising model withDobrushin boundary conditions on it. Since the domain is fixed, set FΎ = FΩ
ÎŽ,aÎŽ,bÎŽ,psd
for the FK fermionic observable.The proof of convergence is in three steps:
First, prove the s-holomorphicity of the observable. Second, prove the convergence of the function HÎŽ naturally associated to thes-holomorphic functions FÎŽ/â2ÎŽ. Third, prove that FÎŽ/â2ÎŽ converges to
âÏâČ.
5.1.1. s-holomorphicity of the (vertex) fermionic observable for FK-Ising. The nexttwo lemmata deal with the edge fermionic observable. They are the key steps of theproof of the s-holomorphicity of the vertex fermionic observable.
Lemma 5.1. For an edge e â ΩΎ , FÎŽ(e) belongs to â(e).Proof. The winding at an edge e can only take its value in the set W +2ÏZ where W
is the winding at e of an arbitrary interface passing through e. Therefore, the windingweight involved in the definition of FÎŽ(e) is always proportional to eiW /2 with a real
coefficient, thus FÎŽ(e) is proportional to eiW /2. In any FK Dobrushin domain, bÎŽ is the
southeast corner and the last edge is thus going to the right. Therefore eiW /2 belongsto â(e) for any e and so does FÎŽ(e). â»
Even though the proof is finished, we make a short parenthetical remark: the defi-nition of s-holomorphicity is not rotationally invariant, nor is the definition of FK Do-brushin domains, since the medial edge pointing to bÎŽ has to be oriented southeast. Thelatter condition has been introduced in such a way that this lemma holds true. Eventhough this condition seems arbitrary, it has no influence on the convergence result,meaning that one could perform a (slightly modified) proof with another orientation.
Lemma 5.2. Consider a medial vertex v in ΩΎ â âΩ
ÎŽ . We have
(5.1) FÎŽ(N) âFÎŽ(S) = i[FÎŽ(E) âFÎŽ(W )]where N , E, S and W are the adjacent edges indexed in clockwise order.
246 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
W
N
E
S
W
N
E
S
v
v
bÎŽ
bÎŽ
Figure 11. Two associated configurations, one with one explorationpath and a loop, one without the loop. One can go from one to theother by switching the state of the edge.
Proof. Let us assume that v corresponds to a primal edge pointing SE to NW , seeFig. 11. The case NE to SW is similar.
We consider the involution s (on the space of configurations) which switches thestate (open or closed) of the edge of the primal lattice corresponding to v. Let e be anedge of the medial graph and set
eÏ â¶= ÏΩΎ,aÎŽ,bÎŽ,psd(Ï) e i
2WÎł(e,bÎŽ)1eâÎł
the contribution of the configuration Ï to FÎŽ(e). Since s is an involution, the followingrelation holds:
FÎŽ(e) =âÏ
eÏ =1
2âÏ
[eÏ + es(Ï)].In order to prove (5.1), it suffices to prove the following for any configuration Ï:
(5.2) NÏ +Ns(Ï) â SÏ â Ss(Ï) = i[EÏ +Es(Ï) âWÏ âWs(Ï)].There are three possibilities:Case 1: the exploration path Îł(Ï) does not go through any of the edges adjacent to v.It is easy to see that neither does Îł(s(Ï)). All the terms then vanish and (5.2) triviallyholds.Case 2: Îł(Ï) goes through two edges around v. Note that it follows the orientation ofthe medial graph, and thus enters v through either W or E and leaves through N or S.We assume that Îł(Ï) enters through the edge W and leaves through the edge S (i.e.that the primal edge corresponding to v is open). The other cases are treated similarly.It is then possible to compute the contributions of all the edges adjacent to v of Ï ands(Ï) in terms of WÏ. Indeed,
The probability of s(Ï) is equal to 1/â2 times the probability of Ï (due to thefact that there is one less open edge of weight 1 â we are at the self-dual pointâ and one less loop of weight
â2, see Proposition 3.17);
CONFORMAL INVARIANCE OF LATTICE MODELS 247
Windings of the curve can be expressed using the winding of W . For instance,the winding of N in the configuration Ï is equal to the winding of W minus aÏ/2 turn.
The contributions are given as:
configuration W E N S
Ï WÏ 0 0 eiÏ/4WÏ
s(Ï) WÏ/â2 eiÏ/2WÏ/â2 eâiÏ/4WÏ/â2 eiÏ/4WÏ/â2Using the identity eiÏ/4 â eâiÏ/4 = i
â2, we deduce (5.2) by summing (with the right
weight) the contributions of all the edges around v.Case 3: Îł(Ï) goes through the four medial edges around v. Then the exploration pathof s(Ï) goes through only two, and the computation is the same as in the second case.
In conclusion, (5.2) is always satisfied and the claim is proved. â»
Recall that the FK fermionic observable is defined on medial edges as well as onmedial vertices. Convergence of the observable means convergence of the vertex observ-able. The edge observable is just a very convenient tool in the proof. The two previousproperties of the edge fermionic observable translate into the following result for thevertex fermionic observable.
Proposition 5.3. The vertex fermionic observable FÎŽ is s-holomorphic.
Proof. Let v be a medial vertex and let N , E, S and W be the four medial edgesaround it. Using Lemmata 5.1 and 5.2, one can see that (5.1) can be rewritten (bytaking the complex conjugate) as:
FÎŽ(N) +FÎŽ(S) = FÎŽ(E) + FÎŽ(W ).In particular, from (3.6),
FÎŽ(v) â¶= 1
2â
e adjacent
FÎŽ(e) = FÎŽ(N) +FÎŽ(S) = FÎŽ(E) +FÎŽ(W ).Using Lemma 5.1 again, FÎŽ(N) and FÎŽ(S) are orthogonal, so that FÎŽ(N) is the pro-jection of FÎŽ(v) on â(N) (and similarly for other edges). Therefore, for a medial edgee = [xy], FÎŽ(e) is the projection of FÎŽ(x) and FÎŽ(y) with respect to â(e), which provesthat the vertex fermionic observable is s-holomorphic. â»
The function FÎŽ/â2ÎŽ is preholomorphic for every ÎŽ > 0. Moreover, Lemma 5.1
identifies the boundary conditions of FÎŽ/â2ÎŽ (its argument is determined) so that thisfunction solves a discrete Riemann-Hilbert boundary value problem. These problemsare significantly harder to handle than the Dirichlet problems. Therefore, it is moreconvenient to work with a discrete analogue of Im (â« z[FÎŽ(z)/â2ÎŽ]2dz), which shouldsolve an approximate Dirichlet problem.
5.1.2. Convergence of (HΎ)Ύ>0. Let A be the black face (vertex of ΩΎ) bordering aΎ,
see Fig. 6. Since the FK fermionic observable FÎŽ/â2ÎŽ is s-holomorphic, Theorem 4.19defines a function HÎŽ ⶠΩΎ âȘΩ
â
ÎŽ â R such that
HÎŽ(A) = 1 and
HÎŽ(B) âHÎŽ(W ) = âŁPâ(e)[FÎŽ(x)]âŁ2 = âŁPâ(e)[FÎŽ(y)]âŁ2for the edge e = [xy] of ΩΎ bordered by a black face B â ΩΎ and a white face W â
ΩâÎŽ . Note that its restriction H to ΩΎ is subharmonic and its restriction HÎŽ to ΩâÎŽ issuperharmonic.
Let us start with two lemmata addressing the question of boundary conditions forHÎŽ.
248 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
B
BâČ
W
e
eâČ
âba
Figure 12. Two adjacent sites B and BâČ on âba together with thenotation needed in the proof of Lemma 5.4.
Lemma 5.4. The function HÎŽ is equal to 1 on the arc âba. The function HÎŽ is equalto 0 on the arc ââab.
Proof. We first prove that HÎŽ is constant on âba. Let B and BâČ be two adjacentconsecutive sites of âba. They are both adjacent to the same dual vertex W â ΩâÎŽ , seeFig. 12. Let e (resp. eâČ) be the edge of the medial lattice between W and B (resp. BâČ).We deduce
HÎŽ (B) âHÎŽ (BâČ) = âŁFÎŽ(e)âŁ2 â âŁFÎŽ(eâČ)âŁ2 = 0(5.3)
The second equality is due to âŁFÎŽ(e)⣠= ÏaÎŽ,bΎΩΎ,psd(W â
â ââab) (see Lemma 7.3). Hence, HÎŽis constant along the arc. Since HÎŽ (A) = 1, the result follows readily.
Similarly, HÎŽ is constant on the arc ââab. Moreover, the dual white face Aâ â ââabbordering aÎŽ (see Fig. 6) satisfies
HÎŽ (Aâ) = HÎŽ (A) â âŁFÎŽ(e)âŁ2 = 1 â 1 = 0(5.4)
where e is the edge separating A and Aâ, which necessarily belongs to Îł. ThereforeHÎŽ = 0 on ââab. â»
Lemma 5.5. The function HÎŽ converges to 0 on the arc âab uniformly away from a
and b, HÎŽ converges to 1 on the arc ââba uniformly away from a and b.
Proof. Once again, we prove the result for HÎŽ . The same reasoning then holds forHÎŽ . Let B be a site of âab at distance r of âba (and therefore at graph distance r/ÎŽ ofâba in ΩΎ). Let W be an adjacent site of B on ââab. Lemma 5.4 implies HÎŽ (W ) = 0.From the definition of HÎŽ, we find
HÎŽ(B) = HÎŽ (W ) + âŁPâ(e)[FÎŽ(e)]âŁ2 = âŁPâ(e)[FÎŽ(e)]âŁ2 = ÏaÎŽ ,bΎΩΎ ,psd
(e â Îł)2.Note that e â Îł if and only if B is connected to the wired arc âba. Therefore, ÏaÎŽ ,bÎŽ
ΩΎ ,psd(e â
Îł) is equal to the probability that there exists an open path from B to âba (the windingis deterministic, see Lemma 7.3 for details). Since the boundary conditions on âab arefree, the comparison between boundary conditions shows that the latter probabilityis smaller than the probability that there exists a path from B to âUÎŽ in the boxUÎŽ = (B + [âr, r]2) â© LÎŽ with wired boundary conditions. Therefore,
HÎŽ (B) = ÏaÎŽ,bΎΩΎ ,psd
(e â Îł)2 †Ï1UÎŽ ,psd(Bâ âUÎŽ)2 .
Proposition 3.10 implies that the right hand side converges to 0 (there is no infinitecluster for Ï1psd,2), which gives a uniform bound for B away from a and b. â»
CONFORMAL INVARIANCE OF LATTICE MODELS 249
The two previous lemmata assert that the boundary conditions for HÎŽ and HÎŽare roughly 0 on the arc âab and 1 on the arc âba. Moreover, HÎŽ and HÎŽ are almostharmonic. This should imply that (HÎŽ)ÎŽ>0 converges to the solution of the Dirichletproblem, which is the subject of the next proposition.
Proposition 5.6. Let (Ω, a, b) be a simply connected domain with two points onthe boundary. Then, (HÎŽ)ÎŽ>0 converges to Im(Ï) uniformly on any compact subsets ofΩ when ÎŽ goes to 0, where Ï is any conformal map from Ω to T = R Ă (0,1) sending ato ââ and b to â.
Before starting, note that Im(Ï) is the solution of the Dirichlet problem on (Ω, a, b)with boundary conditions 1 on âba and 0 on âab.
Proof. From the definition of H , HÎŽ is subharmonic, let hÎŽ be the preharmonicfunction with same boundary conditions as HÎŽ on âΩΎ. Note that HÎŽ †h
ÎŽ. Similarly,hÎŽ is defined to be the preharmonic function with same boundary conditions as HÎŽ onâΩâÎŽ . If K â Ω is fixed, where K is compact, let bÎŽ â KÎŽ and wÎŽ â K
â
ÎŽ any neighbor ofbÎŽ, we have
hΎ(wΎ) †HΎ (wΎ) †HΎ (bΎ) †hΎ(bΎ).(5.5)
Using Lemmata 5.4 and 5.5, boundary conditions forHÎŽ (and therefore hÎŽ) are uniformlyconverging to 0 on âab and 1 on âba away from a and b. Moreover, âŁhÎŽ ⣠is boundedby 1 everywhere. This is sufficient to apply Theorem 4.7: hÎŽ converges to Im(Ï) onany compact subset of Ω when ÎŽ goes to 0. The same reasoning applies to hÎŽ. Theconvergence for HÎŽ and HÎŽ follows easily since they are sandwiched between hÎŽ andhÎŽ. â»
5.1.3. Convergence of FK fermionic observables (FÎŽ/â2ÎŽ)ÎŽ>0. This section contains
the proof of Theorem 3.15. The strategy is straightforward: (FÎŽ/â2ÎŽ)ÎŽ>0 is proved tobe a precompact family for the uniform convergence on compact subsets of Ω. Then,the possible sub-sequential limits are identified using HÎŽ.
Proof of Theorem 3.15. First assume that the precompactness of the family(FÎŽ/â2ÎŽ)ÎŽ>0 has been proved. Let (FÎŽn/â2ÎŽn)nâN be a convergent subsequence anddenote its limit by f . Note that f is holomorphic as it is a limit of preholomorphicfunctions. For two points x, y â Ω, we have:
HÎŽn(y)âHÎŽn(x) = 1
2Im(â« y
x
1
ÎŽnF 2ÎŽn(z)dz)
(for simplicity, also denote the closest points of x, y in ΩΎn by x, y). On the one hand,the convergence of (FÎŽn/â2ÎŽn)nâN being uniform on any compact subset of Ω, the righthand side converges to Im (â« yx f(z)2dz). On the other hand, the left-hand side convergesto Im(Ï(y) â Ï(x)). Since both quantities are holomorphic functions of y, there existsC â R such that Ï(y)âÏ(x) = C +â« yx f(z)2dz for every x, y â Ω. Therefore f equals
âÏâČ.
Since this is true for any converging subsequence, the result follows.Therefore, the proof boils down to the precompactness of (FÎŽ/â2ÎŽ)ÎŽ>0. We will use
the second criterion in Proposition 4.3. Note that it is sufficient to prove this result forsquares Q â Ω such that a bigger square 9Q (with same center) is contained in Ω.
Fix ÎŽ > 0. When jumping diagonally over a medial vertex v, the function HÎŽ changesby Re(F 2
ÎŽ (v)) or Im(F 2ÎŽ (v)) depending on the direction, so that
ÎŽ2 âvâQ
ÎŽ
âŁFÎŽ(v)/â2ÎŽâŁ2 = ÎŽ âxâQÎŽ
âŁâHÎŽ (x)⣠+ ÎŽ âxâQâ
ÎŽ
âŁâHÎŽ (x)âŁ(5.6)
where âHÎŽ(x) = (HÎŽ(x+ ÎŽ)âHÎŽ(x),HÎŽ (x+ iÎŽ)âHÎŽ(x)), and âHÎŽ is defined similarlyfor HÎŽ . It follows that it is enough to prove uniform boundedness of the right hand sidein (5.6). We only treat the sum involving HÎŽ . The other sum can be handled similarly.
250 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
Write HÎŽ = SÎŽ + RÎŽ where SÎŽ is a harmonic function with the same boundaryconditions on â9QÎŽ as HÎŽ . Note that RÎŽ †0 is automatically subharmonic. In order toprove that the sum of âŁâHÎŽ ⣠on QÎŽ is bounded by C/ÎŽ, we deal separately with âŁâSÎŽ âŁand âŁâRÎŽ âŁ. First,
âxâQÎŽ
âŁâSÎŽ(x)⣠†C1
ÎŽ2â C2ÎŽ ( sup
xââQÎŽ
âŁSÎŽ(x)âŁ) †C3
ÎŽ( supxâ9QÎŽ
âŁHÎŽ (x)âŁ) †C4
ÎŽ,
where in the first inequality we used Proposition 4.2 and the maximum principle forSÎŽ, and in the second the fact that SÎŽ and HÎŽ share the same boundary conditions on9QÎŽ. The last inequality comes from the fact that HÎŽ converges, hence remains boundeduniformly in ÎŽ.
Second, recall that G9QÎŽ(â , y) is the Green function in 9QÎŽ with singularity at y.
Since RÎŽ equals 0 on the boundary, Proposition 4.9 implies
RÎŽ(x) = âyâ9QÎŽ
âRÎŽ(y)G9QÎŽ(x, y),(5.7)
thus giving
âRÎŽ(x) = âyâ9QÎŽ
âRÎŽ(y)âxG9QÎŽ(x, y)
Therefore,
âxâQÎŽ
âŁâRÎŽ(x)⣠= âxâQÎŽ
⣠âyâ9QÎŽ
âRÎŽ(y)âxG9QÎŽ(x, y)âŁ
†âyâ9QÎŽ
âRÎŽ(y) âxâQÎŽ
âŁâxG9QÎŽ(x, y)âŁ
†âyâ9QÎŽ
âRÎŽ(y) C5ÎŽ âxâQÎŽ
G9QÎŽ(x, y)
= C5ÎŽ âxâQÎŽ
âyâ9QÎŽ
âRÎŽ(y)G9QÎŽ(x, y)
= C5ÎŽ âxâQÎŽ
RÎŽ(x) = C6/ÎŽThe second line uses the fact that âRÎŽ â„ 0, the third Proposition 4.10, the fifth Propo-sition 4.9 again, and the last inequality the facts that QÎŽ contains of order 1/ÎŽ2 sitesand that RÎŽ is bounded uniformly in ÎŽ (since HÎŽ and SÎŽ are).
Thus, ÎŽâxâQÎŽâŁâHÎŽ ⣠is uniformly bounded. Since the same result holds for HÎŽ ,(FÎŽ/â2ÎŽ)ÎŽ>0 is precompact on Q (and more generally on any compact subset of Ω) and
the proof is completed. â»
5.2. Convergence of the spin fermionic observable. We now turn to the proofof convergence for the spin fermionic observable. Fix a simply connected domain (Ω, a, b)with two points on the boundary. For Ύ > 0, always consider the spin fermionic observableon the discrete spin Dobrushin domain (ΩΎ , aΎ, bΎ). Since the domain is fixed, we set FΎ =FΩ
ÎŽ,aÎŽ,bÎŽ . We follow the same three steps as before, beginning with the s-holomorphicity.
The other two steps are only sketched, since they are more technical than in the FK-Isingcase, see [CS09].
Proposition 5.7. For Ύ > 0, FΎ is s-holomorphic on ΩΎ .
CONFORMAL INVARIANCE OF LATTICE MODELS 251
e
v
x yy
e
v
x yy
e
v
x yy
e
v
x yy
e
v
x ye
y
1(a) 1(b) 3(a) 3(b) 4
e
v
x yy
e
v
x yy
2
e
v
x y
e
v
x ye
ye
v
x y
e
v
x ye
y
e
v
x ye
y
Figure 13. The different possible cases in the proof of Proposition 5.7:Ï is depicted on the top, and ÏâČ on the bottom.
Proof. Let x, y two adjacent medial vertices connected by the edge e = [xy]. Let vbe the vertex of ΩΎ bordering the (medial) edge e. As before, set xÏ (resp. yÏ) for thecontribution of Ï to FÎŽ(x) (resp. FÎŽ(y)). We wish to prove that
âÏ
Pâ(e)(xÏ) = âÏ
Pâ(e)(yÏ).(5.8)
Note that the curve Îł(Ï) finishes at xÏ or at yÏ so that Ï cannot contribute to FÎŽ(x)and FÎŽ(y) at the same time. Thus, it is sufficient to partition the set of configurationsinto pairs of configurations (Ï,ÏâČ), one contributing to y, the other one to x, such thatPâ(e)(xÏ) = Pâ(e)(yÏâČ).
Without loss of generality, assume that e is pointing southeast, thus â(e) = R (othercases can be done similarly). First note that
xÏ =1
Zeâi
1
2[WÎł(Ï)(aÎŽ,xÎŽ)âWÎłâČ(aÎŽ,bÎŽ)](â2 â 1)âŁÏâŁ,
where Îł(Ï) is the interface in the configuration Ï, ÎłâČ is any curve from aÎŽ to bÎŽ (re-call that WÎłâČ(aÎŽ, bÎŽ) does not depend on ÎłâČ), and Z is a normalizing real number notdepending on the configuration. There are six types of pairs that one can create, seeFig. 13 depicting the four main cases. Case 1 corresponds to the case where the interfacereaches x or y and then extends by one step to reach the other vertex. In Case 2, Îłreaches v before x and y, and makes an additional step to x or y. In Case 3, Îł reaches xor y and sees a loop preventing it from being extended to the other vertex (in contrastto Case 1). In Case 4, Îł reaches x or y, then goes away from v and comes back to theother vertex. Recall that the curve must always go to the left: in cases 1(a), 1(b), and2 there can be a loop or even the past of Îł passing through v. However, this does notchange the computation.
We obtain the following table for xÏ and yÏâČ (we always express yÏâČ in terms ofxÏ). Moreover, one can compute the argument modulo Ï of contributions xÏ since theorientation of e is known. When upon projecting on R, the result follows.
configuration Case 1(a) Case 1(b) Case 2 Case 3(a) Case 3(b) Case 4
xÏ xÏ xÏ xÏ xÏ xÏ xÏ
yÏâČ (â2 â 1)eiÏ/4xÏ
eiÏ/4â
2â1xÏ eâiÏ/4xÏ e
3iÏ/4xÏ e3iÏ/4xÏ e
â5iÏ/4xÏ
arg. xÏ mod Ï 5Ï/8 Ï/8 Ï/8 5Ï/8 5Ï/8 5Ï/8
252 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
â»
Proof of Theorem 2.11 (Sketch). The proof is roughly sketched. We refer to [CS09]for a complete proof.
Since FÎŽ is s-harmonic, one can define the observable HÎŽ as in Theorem 4.19, withthe requirement that it is equal to 0 on the white face adjacent to b. Then, HÎŽ is constantequal to 0 on the boundary as in the FK-Ising case. Note that HÎŽ should not convergeto 0, even if boundary conditions are 0 away from a. Firstly, HÎŽ is superharmonic andnot harmonic, even though it is expected to be almost harmonic (away from a, HÎŽ andHÎŽ are close), this will not be true near a. Actually, HÎŽ should not remain boundedaround a.
The main difference compared to the previous section is indeed the unboundednessof HÎŽ near aÎŽ which prevents us from the immediate use of Proposition 4.3. It is actuallypossible to prove that away from a, HÎŽ remains bounded, see [CS09]. This uses moresophisticated tools, among which are the boundary modification trick (see [DCHN10]for a quick description in the FK-Ising case, and [CS09] for the Ising original case). Asbefore, boundedness implies precompactness (and thus boundedness) of (FÎŽ)ÎŽ>0 awayfrom a via Proposition 4.3. Since HÎŽ can be expressed in terms of FÎŽ, it is easy todeduce that HÎŽ is also precompact.
Now consider a convergent subsequence (fÎŽn ,HÎŽn) converging to (f,H). One cancheck that H is equal to 0 on âΩ â a. Moreover, the fact that HÎŽ equals 0 on theboundary and is superharmonic implies that HÎŽ is greater than or equal to 0 everywhere,implying H â„ 0 in Ω. This property of harmonic functions in a domain almost determinesthem. There is only a one-parameter family of positive harmonic functions equal to 0on the boundary. These functions are exactly the imaginary parts of conformal mapsfrom Ω to the upper half-plane H mapping a to â. We can further assume that b ismapped to 0, since we are interested only in the imaginary part of these functions.
Fix one conformal map Ï from Ω to H, mapping a to â and b to 0. There existsλ > 0 such that H = λImÏ. As in the case of the FK-Ising model, one can prove thatIm (â« z f2) =H , implying that f2 = λÏâČ. Since f(b) = 1 (it is obvious from the definition
that FÎŽ(bÎŽ) = 1), λ equals 1ÏâČ(b) . In conclusion, f(z) =âÏâČ(z)/ÏâČ(b) for every z â Ω. â»
Note that some regularity hypotheses on the boundary near b are needed to ensurethat the sequence (fÎŽn ,HÎŽn) also converges near b. This is the reason for assuming thatthe boundary near b is smooth. We also mention that there is no normalization here.The normalization from the point of view of b was already present in the definition ofthe observable.
6. Convergence to chordal SLE(3) and chordal SLE(16/3)
The strategy to prove that a family of parametrized curves converges to SLE(Îș)follows three steps:
First, prove that the family of curves is tight. Then, show that any sub-sequential limit is a time-changed Loewner chain
with a continuous driving process (see Beffaraâs course for details on Loewnerchains and driving processes). Finally, show that the only possible driving processes for the sub-sequential
limits isâÎșBt where Bt is a standard Brownian motion.
The conceptual step is the third one. In order to identify the Brownian motion asbeing the only possible driving process for the curve, we find computable martingalesexpressed in terms of the limiting curve. These martingales will be the limits of fermionicobservables. The fact that these (explicit) functions are martingales allows us to deducemartingale properties of the driving process. More precisely, we aim to use LĂ©vyâs
CONFORMAL INVARIANCE OF LATTICE MODELS 253
Sr,R
γΎ
x
RT
RRRL
RB
[ân, n]2
[â2n, 2n]2
Figure 14. Left: The event A6(x, r,R). In the case of explorationpaths, it implies the existence of alternating open and closed paths.Right: Rectangles RT , RR, RB and RL crossed by closed paths in thelonger direction. The combination of these closed paths prevents theexistence of a crossing from the inner to the outer boundary of theannulus.
theorem: a continuous real-valued process X such that Xt and X2t â at are martingales
is necessarilyâaBt.
6.1. Tightness of interfaces for the FK-Ising model. In this section, we provethe following theorem:
Theorem 6.1. Fix a domain (Ω, a, b). The family (γΎ)Ύ>0 of random interfaces forthe critical FK-Ising model in (Ω, a, b) is tight for the topology associated to the curvedistance.
The question of tightness for curves in the plane has been studied in the ground-breaking paper [AB99]. In that paper, it is proved that a sufficient condition fortightness is the absence, at every scale, of annuli crossed back and forth an unboundednumber of times.
More precisely, for x â Ω and r < R, let Sr,R(x) = (x+ [âR,R]2)â (x + [âr, r]2) anddefine Ak(x; r,R) to be the event that there exist k crossings of the curve γΎ betweenouter and inner boundaries of Sr,R(x).
Theorem 6.2 (Aizenman-Burchard [AB99]). Let Ω be a simply connected domainand let a and b be two marked points on its boundary. Denote by PÎŽ the law of a randomcurve γΎ on ΩΎ from aÎŽ to bÎŽ. If there exist k â N, Ck <â and âk > 2 such that for allÎŽ < r < R and x â Ω,
PÎŽ(Ak(x; r,R)) †Ck( rR)âk
,
then the family of curves (γΎ) is tight.
We now show how to exploit this theorem in order to prove Theorem 6.1. The maintool is Theorem 3.16.
Lemma 6.3 (Circuits in annuli). Let E(x,n,N) be the probability that there existsan open path connecting the boundaries of Sn,N(x). There exists a constant c < 1 suchthat for all n > 0,
Ï1psd,Sn,2n(x)(E(x;n,2n)) †c.Note that the boundary conditions on the boundary of the annulus are wired. Via
comparison between boundary conditions, this implies that the probability of an open
254 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
path from the inner to the outer boundary is bounded uniformly on the configurationoutside of the annulus. This uniform bound allows us to decouple what is happeninginside the annulus with what is happening outside of it.
Proof. Assume x = 0. The result follows from Theorem 3.16 (proved in Section 7.2)applied in the four rectangles RB = [â2n,2n] Ă [ân,â2n], RL = [â2n,ân] Ă [â2n,2n],RT = [â2n,2n]Ă [n,2n] and RR = [n,2n]Ă [â2n,2n], see Fig. 14. Indeed, if there existsa closed path crossing each of these rectangles in the longer direction, one can constructfrom them a closed circuit in Sn,2n. Now, consider any of these rectangles, RB forinstance. Its aspect ratio is 4, so that Theorem 3.16 implies that there is a closed pathcrossing in the longer direction with probability at least c1 > 0 (the wired boundaryconditions are the dual of the free boundary conditions). The FKG inequality (3.2)implies that the probability of a circuit is larger than c41 > 0. Therefore, the probabilityof a crossing is at most c = 1 â c41 < 1. â»
We are now in a position to prove Theorem 6.1.Proof of Theorem 6.1. Fix x â Ω, ÎŽ < r < R and recall that we are on a lattice of
mesh size ÎŽ. Let k to be fixed later. We first prove that
(6.1) ÏaÎŽ ,bΎΩΎ ,psd
(A2k(x; r,2r)) †ckfor some constant c < 1 uniform in x, k, r, Ύ and the configuration outside of Sr,2r(x).
If A2k(x; r,2r) holds, then there are (at least) k open paths, alternating with k dualpaths, connecting the inner boundary of the annulus to its outer boundary. Since thepaths are alternating, one can deduce that there are k open crossings, each one beingsurrounded by closed crossings. Hence, using successive conditionings and the compar-ison between boundary conditions, the probability for each crossing is smaller than theprobability that there is a crossing in the annulus with wired boundary conditions (sincethese boundary conditions maximize the probability of E(x; r,2r)). We obtain
ÏaÎŽ,bΎΩΎ ,psd
(A2k(x; r,2r)) †[Ï1psd,Sr,2r(x)(E(x; r,2r))]k .Using Lemma 6.3, Ï1
psd,Sr,2r(x)(E(x; r,2r)) †c < 1 and and (6.1) follows.
One can further fix k large enough so that ck < 18. Now, one can decompose the
annulus Sr,R(x) into roughly ln2(R/r) annuli of the form Sr,2r(x), so that for theprevious k,
(6.2) ÏaÎŽ,bΎΩΎ,psd
(A2k(x; r,R)) †( rR)3 .
Hence, Theorem 6.2 implies that the family (γΎ) is tight. â»
6.2. sub-sequential limits of FK-Ising interfaces are Loewner chains. Thissubsection requires basic knowledge of Loewner chains and we refer to Beffaraâs coursein this volume for an overview on the subject. In the previous subsection, traces of inter-faces in Dobrushin domains were shown to be tight. The natural discrete parametriza-tion does not lead to a suitable continuous parametrization. We would prefer our sub-sequential limits to be parametrized as Loewner chains. In other words, we would liketo parametrize the curve by its so-called h-capacity. In this case, we say that the curveis a time-changed Loewner chain.
Theorem 6.4. Any sub-sequential limit of the family (γΎ)Ύ>0 of FK-Ising interfacesis a time-changed Loewner chain.
CONFORMAL INVARIANCE OF LATTICE MODELS 255
b
Sr,R
a
b
a
b
Figure 15. Left: An example of a fjord. Seen from b, the h-capacity(roughly speaking, the size) of the hull does not grow much while thecurve is in the fjord. The event involves six alternating open and closedcrossings of the annulus. Right: Conditionally on the beginning of thecurve, the crossing of the annulus is unforced on the left, while it isforced on the right (it must go ultimately to b).
Not every continuous curve is a time-changed Loewner chain. In the case of FKinterfaces, the limiting curve is fractal-like and has many double points, so that the fol-lowing theorem is not a trivial statement. A general characterization for a parametrizednon-selfcrossing curve in (Ω, a, b) to be a time-changed Loewner chain is the following:
its h-capacity must be continuous, its h-capacity must be strictly increasing. the curve grows locally seen from infinity in the following sense: for any t â„ 0
and for any Δ > 0, there exists ÎŽ > 0 such that for any s †t, the diameterof gs(Ωs â Ωs+ÎŽ) is smaller than Δ, where Ωs is the connected component ofΩâÎł[0, s] containing b and gs is the conformal map from Ωs to H with hydro-dynamical renormalization (see Beffaraâs course).
The first condition is automatically satisfied by continuous curves. The third one usuallyfollows from the two others when the curve is continuous, so that the crucial conditionto check is the second one. This condition can be understood as being the fact that thetip of the curve is visible from b at every time. In other words, the family of hulls createdby the curve (i.e. the complement of the connected component of Ω â Îłt containing b)is strictly increasing. This is the case if the curve does not enter long fjords created byits past at every scale, see Fig. 15.
In the case of FK interfaces, this corresponds to so-called six arm event, and itboils down to proving that â6 > 2. A general belief in statistical physics is that manyexponents, called universal exponents, do not depend on the model. For instance, theso-called 5-arm exponent should equal 2. This would imply that â6 > â5 = 2. Ingeneral, proving that the 5-arm exponent equals 2 is very hard. Therefore, we need toinvoke a stronger structural theorem to prove that sub-sequential limits are Loewnerchains. Recently, Kemppainen and the second author proved the required theorem, andwe describe it now.
For a family of parametrized curves (γΎ)ÎŽ>0, define Condition (â) by:
Condition (â): There exist C > 1 and â > 0 such that for any 0 < ÎŽ < r < R/C, forany stopping time Ï and for any annulus Sr,R(x) not containing ÎłÏ , the probability thatγΎ crosses the annulus Sr,R(x) (from the outside to the inside) after time Ï while it is
not forced to enter Sr,R(x) again is smaller than C(r/R)â, see Fig. 15.
Roughly speaking, the previous condition is a uniform bound on unforced crossings.Note that it is necessary to assume the fact that the crossing is unforced.
256 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
Theorem 6.5 ([KS10]). If a family of curves (γΎ) satisfies Condition (â), then itis tight for the topology associated to the curve distance. Moreover, any sub-sequentiallimit (γΎn) is to a time-changed Loewner chain.
Tightness is almost obvious, since Condition (â) implies the hypothesis in Aizenman-Burchardâs theorem. The hard part is the proof that Condition (â) guarantees that theh-capacity of sub-sequential limits is strictly increasing and that they create Loewnerchains. The reader is referred to [KS10] for a proof of this statement. We are now in aposition to prove Theorem 6.4:
Proof of Theorem 6.4. Lemma 6.3 allows us to prove Condition (â) without difficulty.â»
6.3. Convergence of FK-Ising interfaces to SLE(16/3). The FK fermionicobservable is now proved to be a martingale for the discrete curves and to identify thedriving process of any sub-sequential limit of FK-Ising interfaces.
Lemma 6.6. Let ÎŽ > 0. The FK fermionic observable M ÎŽn(z) = FΩΎâÎł[0,n],Îłn,bÎŽ(z)
is a martingale with respect to (Fn), where Fn is the Ï-algebra generated by the FKinterface Îł[0, n].
Proof. For a Dobrushin domain (ΩΎ , aΎ, bΎ), the slit domain created by "removing"the first n steps of the exploration path is again a Dobrushin domain. Conditionally on
Îł[0, n], the law of the FK-Ising model in this new domain is exactly ÏÎłn,bΎΩ
ÎŽâÎł[0,n]. This
observation implies that M ÎŽn(z) is the random variable 1zâγΎe
1
2iWγΎ
(z,b) conditionally onFn, therefore it is automatically a martingale. â»
Proposition 6.7. Any sub-sequential limit of (γΎ)ÎŽ>0 which is a Loewner chain isthe (chordal) Schramm-Loewner Evolution with parameter Îș = 16/3.
Proof. Consider a sub-sequential limit Îł in the domain (Ω, a, b) which is a Loewnerchain. Let Ï be a map from (Ω, a, b) to (H,0,â). Our goal is to prove that Îł = Ï(Îł) isa chordal SLE(16/3) in the upper half-plane.
Since Îł is assumed to be a Loewner chain, Îł is a growing hull from 0 to âparametrized by its h-capacity. Let Wt be its continuous driving process. Also, de-fine gt to be the conformal map from Hâ Îł[0, t] to H such that gt(z) = z+2t/z+O(1/z2)when z goes to â.
Fix zâČ â Ω. For ÎŽ > 0, recall that M ÎŽn(zâČ) is a martingale for γΎ. Since the martingale
is bounded, M ÎŽÏt(zâČ) is a martingale with respect to FÏt , where Ït is the first time
at which Ï(γΎ) has an h-capacity larger than t. Since the convergence is uniform,Mt(zâČ) â¶= limÎŽâ0M
ÎŽÏt(zâČ) is a martingale with respect to Gt, where Gt is the Ï-algebra
generated by the curve Îł up to the first time its h-capacity exceeds t. By definition, thistime is t, and Gt is the Ï-algebra generated by Îł[0, t].
Recall that Mt(zâČ) is related to Ï(zâČ) via the conformal map from H â Îł[0, t] toRĂ(0,1), normalized to send Îłt to ââ andâ toâ. This last map is exactly 1
Ïln(gtâWt).
Setting z = Ï(zâČ), we obtain that
âÏMz
t â¶=âÏMt(zâČ) = â[ln(gt(z)âWt)]âČ =
ÂżĂĂĂ gâČt(z)gt(z)âWt
(6.3)
is a martingale. Recall that, when z goes to infinity,
gt(z) = z +2t
z+O ( 1
z2) and gâČt(z) = 1 â
2t
z2+O ( 1
z3)(6.4)
CONFORMAL INVARIANCE OF LATTICE MODELS 257
For s †t,
âÏ â E[Mz
t âŁGs] = E
âĄâąâąâąâąâŁÂżĂĂĂ 1 â 2t/z2 +O(1/z3)
z âWt + 2t/z +O(1/z2) ⣠Gsâ€â„â„â„â„âŠ
=1âzE [1 + 1
2Wt/z + 1
8(3W 2
t â 16t) /z2 +O (1/z3) ⣠Gs]=
1âz(1 + 1
2E[WtâŁGs]/z + 1
8E[3W 2
t â 16tâŁGs]/z2 +O (1/z3)) .Taking s = t yields
âÏ â Mz
s =1âz(1 + 1
2Ws/z + 1
8(3W 2
s â 16s)/z2 +O(1/z3)) .Since E[Mz
t âŁGs] = Mzs , terms in the previous asymptotic development can be matched
together so that E[WtâŁGs] =Ws and E[W 2t â
163tâŁGs] =W 2
s â163s. Since Wt is continuous,
LĂ©vyâs theorem implies that Wt =
â163Bt where Bt is a standard Brownian motion.
In conclusion, Îł is the image by Ïâ1 of the chordal Schramm-Loewner Evolutionwith parameter Îș = 16/3 in the upper half-plane. This is exactly the definition of thechordal Schramm-Loewner Evolution with parameter Îș = 16/3 in the domain (Ω, a, b).â»
Proof of Theorem 3.13. By Theorem 4.3, the family of curves is tight. Using The-orem 6.4, any sub-sequential limit is a time-changed Loewner chain. Consider such asub-sequential limit and parametrize it by its h-capacity. Proposition 6.7 then impliesthat it is the Schramm-Loewner Evolution with parameter Îș = 16/3. The possible limitbeing unique, the claim is proved. â»
6.4. Convergence to SLE(3) for spin Ising interfaces. The proof of Theo-rem 2.10 is very similar to the proof of Theorem 3.13, except that we work with the spinIsing fermionic observable instead of the FK-Ising model one. The only point differingfrom the previous section is the proof that the spin fermionic observable is a martingalefor the curve. We prove this fact now and leave the remainder of the proof as an exercise.Let Îł be the interface in the critical Ising model with Dobrushin boundary conditions.
Lemma 6.8. Let Ύ > 0, the spin fermionic observable M Ύn(z) = FΩ
ÎŽâÎł[0,n],Îł(n),bÎŽ(z) is
a martingale with respect to (Fn), where Fn is the Ï-algebra generated by the explorationprocess Îł[0, n].
Proof. It is sufficient to check that FÎŽ(z) has the martingale property when Îł = Îł(Ï)makes one step Îł1. In this case F0 is the trivial Ï-algebra, so that we wish to prove
”a,bÎČc,Ω[FΩ
ÎŽâ[aÎŽÎł1],Îł1,bÎŽ(z)] = FΩ
ÎŽ,aÎŽ,bÎŽ(z),(6.5)
where ”a,bÎČc,Ω
is the critical Ising measure with Dobrushin boundary conditions in Ω.
Write ZΩΎ,aÎŽ,bÎŽ (resp. ZΩâ[aÎŽx],x,bÎŽ) for the partition function of the Ising model
with Dobrushin boundary conditions on (ΩΎ , aÎŽ, bÎŽ) (resp. (Ω â [aÎŽx], x, bÎŽ)), i.e.
ZΩâ[aÎŽx],x,bÎŽ = âÏ(â2 â 1)âŁÏâŁ. Note that ZΩâ[aÎŽx],x,bÎŽ is almost the denominator of
258 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
FΩΎâ[aÎŽx],x,bÎŽ(zÎŽ). By definition,
ZΩΎ,aΎ,bΎ ”
a,bÎČc,Ω(Îł1 = x) = (â2 â 1)ZΩâ[aÎŽx],x,bÎŽ
= (â2 â 1)ei 12WÎł(x,bÎŽ)âÏâE
Ωâ[aÎŽx](x,zÎŽ) eâi 1
2WÎł(x,zÎŽ)(â2 â 1)âŁÏâŁ
FΩΎâ[aÎŽx],x,bÎŽ(zÎŽ)
= ei1
2WÎł(aÎŽ,bÎŽ)
âÏâEΩΎ(aÎŽ,zÎŽ) e
âi 12WÎł(aÎŽ,zÎŽ)(â2 â 1)âŁÏâŁ1Îł1=x
FΩΎâ[aÎŽx],x,bÎŽ(zÎŽ)
In the second equality, we used the fact that EΩΎâ[aÎŽx](x, zÎŽ) is in bijection with config-
urations of EΩΎ(aΎ, zΎ) such that γ1 = x (there is still a difference of weight of
â2 â 1
between two associated configurations). This gives
”a,bÎČc,Ω(Îł1 = x)FΩ
ÎŽâ[aÎŽx],x,bÎŽ(zÎŽ) = âÏâE(aÎŽ,zÎŽ) eâi
1
2WÎł(aÎŽ,zÎŽ)(â2 â 1)âŁÏâŁ1Îł1=x
eâi1
2Wγ(aΎ,bΎ)ZΩ
ÎŽ,aÎŽ,bÎŽ
.
The same holds for all possible first steps. Summing over all possibilities, we obtain theexpectation on one side of the equality and FΩ
ÎŽ,aÎŽ,bÎŽ(zÎŽ) on the other side, thus proving
(6.5). â»
Exercise 6.9. Prove that spin Ising interfaces converge to SLE(3). For tightnessand the fact that sub-sequential limits are Loewner chains, it is sufficient to check Con-dition (â). To do so, try to use Theorem 3.16 and the Edwards-Sokal coupling to provean intermediate result similar to Lemma 6.3.
7. Other results on the Ising and FK-Ising models
7.1. Massive harmonicity away from criticality. In this subsection, we con-sider the fermionic observable F for the FK-Ising model away from criticality. The Isingmodel is still solvable and the observable becomes massive harmonic (i.e. âf = λ2f).We refer to [BDC11] for details on this paragraph. We start with a lemma which
extends Lemma 5.2 to p â psd =â2/(1 +â2).
Lemma 7.1. Let p â (0,1). Consider a vertex v â Ω â âΩ,
(7.1) F (A) âF (C) = ieiα [F (B) âF (D)]where A is an adjacent (to v) medial edge pointing towards v and B, C and D areindexed in such a way that A, B, C and D are found in counterclockwise order. Theparameter α is defined by
eiα =eâiÏ/4(1 â p)â2 + peâiÏ/4p + (1 â p)â2 .
The proof of this statement follows along the same lines as the proof of Lemma 5.2.
Proposition 7.2. For p <â2/(1+â2), there exists Ο = Ο(p) > 0 such that for every
n,
Ïp(0â in) †eâΟn,(7.2)
where the mesh size of the lattice L is 1.
In this proof, the lattices are rotated by an angle Ï/4. We will be able to estimatethe connectivity probabilities using the FK fermionic observable. Indeed, the observableon the free boundary is related to the probability that sites are connected to the wiredarc. More precisely:
CONFORMAL INVARIANCE OF LATTICE MODELS 259
Lemma 7.3. Fix (G,a, b) a Dobrushin domain and p â (0,1). Let u â G be a site onthe free arc, and e be a side of the black diamond associated to u which borders a whitediamond of the free arc. Then,
(7.3) âŁF (e)⣠= Ïa,bp,G(uâ wired arc).
Proof. Let u be a site of the free arc and recall that the exploration path is theinterface between the open cluster connected to the wired arc and the dual open clusterconnected to the free arc. Since u belongs to the free arc, u is connected to the wiredarc if and only if e is on the exploration path, so that
Ïa,bp,G(uâ wired arc) = Ïa,bp,G(e â Îł).
The edge e being on the boundary, the exploration path cannot wind around it, so thatthe winding (denoted W1) of the curve is deterministic (and easy to write in terms ofthat of the boundary itself). We deduce from this remark that
âŁF (e)⣠= âŁÏa,bp,G(e i
2W11eâÎł)⣠= âŁe i
2W1Ï
a,bp,G(e â Îł)âŁ
= Ïa,bp,G(e â Îł) = Ïa,bp,G(uâ wired arc).
â»
We are now in a position to prove Proposition 7.2. We first prove exponential decayin a strip with Dobrushin boundary conditions, using the observable. Then, we useclassical arguments of FK percolation to deduce exponential decay in the bulk. Wepresent the proof quickly (see [BDC11] for a complete proof).
Proof. Let p < psd, and consider the FK-Ising model of parameter p in the strip ofheight â, with free boundary conditions on the top and wired boundary conditions onthe bottom (the measure is denoted by Ïâ,ââp,Sâ
). It is easy to check that one can definethe FK fermionic observable F in this case, by using the unique interface from ââ toâ. This observable is the limit of finite volume observables, therefore it also satisfiesLemma 7.1.
Let ek be the medial edge with center ik + 1+iâ2, see Fig. 16. A simple computation
using Lemmata 7.1 and 5.1 plus symmetries of the strip (via translation and horizontalreflection) implies
F (ek+1) = [1 + cos(Ï/4 â α)] cos(Ï/4 â α)[1 + cos(Ï/4 + α)] cos(Ï/4 + α)F (ek) .(7.4)
Using the previous equality inductively, we find for every â > 0,
âŁF (eâ)⣠= eâΟââŁF (e0)⣠†eâΟâwith
Ο â¶= â ln[1 + cos(Ï/4 â α)] cos(Ï/4 â α)[1 + cos(Ï/4 + α)] cos(Ï/4 + α) .(7.5)
Since eâ is adjacent to the free arc, Lemma 7.3 implies
Ïâ,ââp,Sâ
[iââ Z] = âŁF (eâ)⣠†eâΟâNow, let N â N and recall that Ï0p,N â¶= Ï
0p,2,[âN,N]2 converges to the infinite-volume
measure with free boundary conditions Ï0p when N goes to infinity.
Consider a configuration in the box [âN,N]2, and let Amax be the site of the clusterof the origin which maximizes the ââ-norm maxâŁx1âŁ, âŁx2 ⣠(it could be equal to N). Ifthere is more than one such site, we consider the greatest one in lexicographical order.Assume that Amax equals a = a1 + ia2 with a2 â„ âŁa1⣠(the other cases can be treated thesame way by symmetry, using the rotational invariance of the lattice).
260 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
ek
ek+1
(0, k + 1)
(0, k)
a = (a1, a2)
explored
L
Îł
area
Figure 16. Left: Edges ek and ek+1. Right: A dual circuit surround-ing an open path in the box [âa2, a2]2. Conditioning on to the mostexterior such circuit gives no information on the state of the edges insideit.
By definition, if Amax equals a, a is connected to 0 in [âa2, a2]2. In addition to this,because of our choice of the free boundary conditions, there exists a dual circuit startingfrom a + i/2 in the dual of [âa2, a2]2 (which is the same as L
â â© [âa2 â 1/2, a2 + 1/2]2)and surrounding both a and 0. Let Î be the outermost such dual circuit: we get
(7.6) Ï0p,N(Amax = a) =âÎł
Ï0p,N(aâ 0âŁÎ = Îł)Ï0p,N(Î = Îł),where the sum is over contours Îł in the dual of [âa2, a2]2 that surround both a and 0.
The event Î = Îł is measurable in terms of edges outside or on Îł. In addition,conditioning on this event implies that the edges of Îł are dual-open. Therefore, from thedomain Markov property, the conditional distribution of the configuration inside Îł is aFK percolation model with free boundary conditions. Comparison between boundaryconditions implies that the probability of a â 0 conditionally on Î = Îł is smallerthan the probability of a â 0 in the strip Sa2 with free boundary conditions on thetop and wired boundary conditions on the bottom. Hence, for any such Îł, we get
Ï0p,N (aâ 0âŁÎ = Îł) †Ïâ,ââp,Sa2
(aâ 0) = Ïâ,ââp,Sa2
(aâ Z) †eâΟa2(observe that for the second measure, Z is wired, so that aâ 0 and aâ Z have thesame probability). Plugging this into (7.6), we obtain
Ï0p,N(Amax = a) â€âÎł
eâΟmaxa1,a2 Ï0p,N(Î = Îł) †eâΟa2 = eâΟmaxa1,a2.
Fix n †N . We deduce from the previous inequality that there exists a constant0 < c <â such that
Ï0p,N(0â Z2â [ân,n]2) †â
aâ[âN,N]2â[ân,n]2Ï0p,N(Amax = a) †cneâΟn.
Since the estimate is uniform in N , we deduce that
(7.7) Ï0p(0â in) †Ï0p(0â Z2â [ân,n]2) †cneâΟn.
â»
Theorem 7.4. The critical parameter for the FK-Ising model isâ2/(1+â2). The
critical inverse-temperature for the Ising model is 12ln(1 +â2).
CONFORMAL INVARIANCE OF LATTICE MODELS 261
Proof. The inequality pc â„â2/(1 +â2) follows from Proposition 7.2 since there is
no infinite cluster for Ï0p,2 when p < psd (the probability that 0 and in are connected
converges to 0). In order to prove that pc â€â2/(1 + â2), we harness the following
standard reasoning.Let An be the event that the point n â N is in an open circuit which surrounds
the origin. Notice that this event is included in the event that the point n â N is in acluster of radius larger than n. For p <
â2/(1+â2), a modification of (7.7) implies that
the probability of An decays exponentially fast. The Borel-Cantelli lemma shows thatthere is almost surely a finite number of n such that An occurs. In other words, thereis a.s. only a finite number of open circuits surrounding the origin, which enforces theexistence of an infinite dual cluster whenever p <
â2/(1+â2). Using duality, the primal
model is supercritical whenever p >â2/(1 +â2), which implies pc â€
â2/(1 +â2). â»
In fact, the FK fermionic observable FÎŽ in a Dobrushin domain (ΩΎ , aÎŽ, bÎŽ) is massiveharmonic when p â psd. More precisely,
Proposition 7.5. Let p â psd,
âÎŽFÎŽ(v) = (cos2α â 1)FÎŽ(v)(7.8)
for every v â ΩΎ ââΩ
ÎŽ , where âÎŽ is the average on sites at distanceâ2ÎŽ minus the value
at the point.
When ÎŽ goes to 0, one can perform two scaling limits. If p = psd(1âλΎ) goes to psd asÎŽ goes to 0, 1
ÎŽ2(âÎŽ + [1â cos 2α]I) converges to â+λ2I. Then FÎŽ (properly normalized)
should converge to a function f satisfying âf + λ2f = 0 inside the domain. Except forλ = 0, the limit will not be holomorphic, but massive harmonic. Discrete curves shouldconverge to a limit which is not absolutely continuous with respect to SLE(16/3). Thestudy of this regime, connected to massive SLEs, is a very interesting subject.
If we fix p < psd, one can interpret massive harmonicity in terms of killed randomwalks. Roughly speaking, FÎŽ(v) is the probability that a killed random walk starting atv visits the wired arc âba. Large deviation estimates on random walks allow to computethe asymptotic of FÎŽ inside the domain. In [BDC11], a surprising link (first noticed byMessikh [Mes06]) between correlation lengths of the Ising model and large deviationsestimates of random walks is presented. We state the result in the following theorem:
Theorem 7.6. Fix ÎČ < ÎČc (and α associated to it) and set
m(ÎČ) â¶= cos(2α).For any x â L,
(7.9) â limnââ
1
nln”ÎČ[Ï0Ï(nx)] = â lim
nââ
1
nlnGm(ÎČ)(0, nx).
Above, Gm(0, x) â¶= Ex[mÏ ] for any x â L and m < 1, where Ï is the hitting time of theorigin and P
x is the law of a simple random walk starting at x.
The massive Green function Gm(0, x) on the right of (7.9) has been widely studied.In particular, we can compute the rate of decay in any direction and deduce Theorem 2.3and Theorem 2.4 (see e.g. [Mes06]).
Exercise 7.7. Prove Lemma 7.1 and the fact that F is massive harmonic insidethe domain.
262 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
7.2. Russo-Seymour-Welsh Theorem for FK-Ising. In this section, we sketchthe proof of Theorem 3.16; see [DCHN10] for details. This theorem was improved in[CDH12]. We would like to emphasize that this result does not make use of scalinglimits. Therefore, it is mostly independent of Sections 5 and 6.
We start by presenting a link between discrete harmonic measures and the proba-bility for a point on the free arc âab of a FK Dobrushin domain to be connected to thewired arc âba.
Let us first define a notion of discrete harmonic measure in a FK Dobrushin domainΩΎ which is slightly different from the usual one. First extend ΩΎ âȘΩ
â
ÎŽ by adding twoextra layers of vertices: one layer of white faces adjacent to ââab, and one layer of black
faces adjacent to âba. We denote the extended domains by ΩΎ and ΩâÎŽ .Define (Xt )tâ„0 to be the continuous-time random walk on the black faces that jumps
with rate 1 on neighbors, except for the faces on the extra layer adjacent to âab ontowhich it jumps with rate Ï â¶= 2/(â2+1). For B â ΩΎ, we denote by H(B) the probability
that the random walk Xt starting at B hits âΩΎ on the wired arc âba (in other words, ifthe random walk hits âba before hitting the extra layer adjacent to ââab). This quantityis called the (modified) harmonic measure of âba seen from B. Similarly, one can definea modified random walk Xt and the associated harmonic measure of ââab seen from w.We denote it by H(w).
Proposition 7.8. Consider a FK Dobrushin domain (ΩΎ, aÎŽ, bÎŽ), for any site B onthe free arc âab,
(7.10)âH(W ) †Ï
aΎ,bΎΩΎ,psd
[Bâ âba] †âH(B),where W is any dual neighbor of B not on ââab.
This proposition raises a connection between harmonic measure and connectivityproperties of the FK-Ising model. To study connectivity probabilities for the FK-Isingmodel, it suffices to estimate events for simple random walks (slightly repelled on theboundary). The proof makes use of a variant of the "boundary modification trick".This trick was introduced in [CS09] to prove Theorem 2.11. It can be summarized asfollows: one can extend the function H by 0 or 1 on the two extra layers, then H (resp.H) is subharmonic (resp. superharmonic) for the Laplacian associated to the randomwalk X (resp. X). Interestingly, H is not subharmonic for the usual Laplacian. Thistrick allows us to fix the boundary conditions (0 or 1), at the cost of a slightly modifiednotion of harmonicity.
We can now give the idea of the proof of Theorem 3.16 (we refer to [DCHN10] fordetails). The proof is a second moment estimate on the number of pairs of connectedsites on opposite edges of the rectangle. We mention that another road to Theorem 3.16has been proposed in [KS10].
Proof of Theorem 3.16 (Sketch). Let Rn = [0,4n]Ă[0, n] be a rectangle and let N bethe number of pairs (x, y), x â 0Ă [0, n] and y â 4nĂ [0, n] such that x is connectedto y by an open path. The expectation of N is easy to obtain using the previousproposition. Indeed, it is the sum over all pairs x, y of the probability of xâ y whenthe boundary conditions are free. Free boundary conditions can be thought of as adegenerate case of a Dobrushin domain, where a = b = y. In other words, we want toestimate the probability that x is connected to the wired arc âba = y. Except when xand y are close to the corners, the harmonic measure of y seen from x is of order 1/n2,so that the probability of x â y is of order 1/n. Therefore, there exists a universal
constant c > 0 such that Ïfpsd,Rn[N] â„ cn.
The second moment estimate is harder to obtain, as usual. Nevertheless, it can be
proved, using successive conditioning and Proposition 7.8, that Ïfpsd,Rn[N2] †Cn2 for
some universal C > 0; see [DCHN10] for a complete proof. Using the Cauchy-Schwarz
CONFORMAL INVARIANCE OF LATTICE MODELS 263
inequality, we find
Ïfpsd,Rn
[N > 0] Ïfpsd,Rn[N2] â„ Ï
fpsd,Rn
[N]2(7.11)
which implies
ÏfRn,psd
[â open crossing] = ÏfRn,psd
[N > 0] â„ c2/C(7.12)
uniformly in n. â»
We have already seen that Theorem 3.16 is central for proving tightness of interfaces.We would also like to mention an elementary consequence of Theorem 3.16.
Proposition 7.9. There exist constants 0 < c,C, ÎŽ,â < â such that for any sitesx, y â L,
câŁx â yâŁÎŽ †”ÎČc[ÏxÏy] †CâŁx â yâŁâ(7.13)
where ”ÎČcis the unique infinite-volume measure at criticality.
Proof. Using the Edwards-Sokal coupling, (7.13) can be rephrased as
câŁx â yâŁÎŽ †Ïpsd,2[xâ y] †CâŁx â yâŁâ ,where Ïpsd,2 is the unique FK-Ising infinite-volume measure at criticality. In order toget the upper bound, it suffices to prove that Ïpsd,2(0 â âÎk) decays polynomiallyfast, where Îk is the box of size k = âŁx â y⣠centered at x. We consider the annuliAn = S2nâ1,2n(x) for n †ln2 k, and E(An) the event that there is an open path crossingAn from the inner to the outer boundary. We know from Corollary 6.3 (which is a directapplication of Theorem 3.16) that there exists a constant c < 1 such that
Ï1An,psd,2(E(An)) †c
for all n â„ 1. By successive conditionings, we then obtain
Ïpsd,2(0â âÎk) †ln2 kân=1
Ï1An,psd,2(E(An)) †cN ,
and the desired result follows. The lower bound can be done following the same kind ofarguments (we leave it as an exercise). â»
Therefore, the behavior at criticality (power law decay of correlations) is very differ-ent from the subcritical phase (exponential decay of correlations). Actually, the previousresult is far from optimal. One can compute correlations between spins of a domain veryexplicitly. In particular, ”ÎČc
[ÏxÏy] behaves like âŁx â yâŁâα, where α = 1/4. We mentionthat α is one example of critical exponent. Even though we did not discuss how computecritical exponents, we mention that the technology developed in these notes has for itsmain purpose their computation.
To conclude this section, we mention that Theorem 3.16 leads to ratio mixing prop-erties (see Exercise 7.10) of the Ising model. Recently, Lubetzky and Sly [LS10] usedthese spatial mixing properties in order to prove an important conjecture on the mixingtime of the Glauber dynamics of the Ising model at criticality.
Exercise 7.10 (Spatial mixing). Prove that there exist c,â > 0 such that for anyr †R,
(7.14) âŁÏpsd,2(A â©B) â Ïpsd,2(A)Ïpsd,2(B)⣠†c( rR)â
Ïpsd,2(A)Ïpsd,2(B)for any event A (resp. B) depending only on the edges in the box [âr, r]2 (resp. outside[âR,R]2).
264 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
7.3. Discrete singularities and energy density of the Ising model. In thissubsection, we would like to emphasize the fact that slight modifications of the spinfermionic observable can be used directly to compute interesting quantities of the model.Now, we briefly present the example of the energy density between two neighboring sitesx and y (Theorem 2.12).
So far, we considered observables depending on a point a on the boundary of adomain, but we could allow more flexibility and move a inside the domain: for aÎŽ â Ω
ÎŽ ,we define the fermionic observable F aÎŽ
ΩΎ(zÎŽ) for zÎŽ â aÎŽ by
F aΎΩΎ(zΎ) = λ
âÏâE(aÎŽ,zÎŽ) eâ 1
2iWÎł(Ï)(aÎŽ,zÎŽ)(â2 â 1)âŁÏâŁ
âÏâE(â2 â 1)âŁÏâŁ(7.15)
where λ is a well-chosen explicit complex number. Note that the denominator of the
observable is simply the partition function for free boundary conditions ZfÎČc,G
. Actually,
using the high-temperature expansion of the Ising model and local rearrangements, theobservable can be related to spin correlations [HS10]:
Lemma 7.11. Let [xy] be an horizontal edge of ΩΎ. Then
λ ”fÎČc,ΩΎ
[ÏxÏy] = Pâ(ac)[F aΩΎ(c)] +Pâ(ad)[F aΩΎ
(d)]where a is the center of [xy], c = a + ÎŽ 1+iâ
2and d = a â ÎŽ 1+iâ
2.
If λ is chosen carefully, the function F aΎΎ
is s-holomorphic on ΩΎ â aÎŽ. Moreover,its complex argument is fixed on the boundary of the domain. Yet, the function isnot s-holomorphic at aÎŽ (meaning that there is no way of defining F aΩΎ
(aÎŽ) so that thefunction is s-holomorphic at aÎŽ). In other words, there is a discrete singularity at a,whose behavior is related to the spin-correlation.
We briefly explain how one can address the problem of discrete singularities, andwe refer to [HS10] for a complete study of this case. In the continuum, singularities areremoved by subtracting Green functions. In the discrete context, we will do the same.We thus need to construct a discrete s-holomorphic Green function. PreholomorphicGreen functions6 were already constructed in [Ken00]. These functions are not s-holomorphic but relevant linear combinations of them are, see [HS10]. We mentionthat the s-holomorphic Green functions are very explicit and their convergence whenthe mesh size goes to 0 can be studied.
Proof of Theorem 2.12 (Sketch). The function F aÎŽÎŽ/ÎŽ converges uniformly on any
compact subset of Ω â a. This fact is not helpful, since the interesting values of F aÎŽÎŽ
are located at neighbors of the singularity. It can be proved that, subtracting a well-chosen s-holomorphic Green function gaΎΩΎ
, one can erase the singularity at aÎŽ. More
precisely, one can show that [F aÎŽÎŽâ gaΎΩΎ
]/Ύ converges uniformly on Ω towards an explicit
conformal map. The value of this map at a is λÏÏâČa(a). Now, ”f
ÎČc,ΩΎ[ÏxÏy] can be
expressed in terms of F aÎŽÎŽ
for neighboring vertices of aΎ. Moreover, values of gaΎΩΎfor
neighbors of aÎŽ can be computed explicitly. Using the fact that
F aÎŽÎŽ= gaÎŽ
ΩΎ+ ÎŽ â
1
ÎŽ[F aÎŽÎŽâ gaÎŽ
ΩΎ],
and Lemma 7.11, the convergence result described above translates into the followingasymptotics for the spin correlation of two neighbors
”fÎČc,ΩΎ
[ÏxÏy] =â2
2â ÎŽ
1
ÏÏâČa(a) + o(ÎŽ).
â»
6i.e. satisfying the Cauchy-Riemann equation except at a certain point.
CONFORMAL INVARIANCE OF LATTICE MODELS 265
8. Many questions and a few answers
8.1. Universality of the Ising model. Until now, we considered only the squarelattice Ising model. Nevertheless, normalization group theory predicts that the scalinglimit should be universal. In other words, the limit of critical Ising models on planargraphs should always be the same. In particular, the scaling limit of interfaces in spinDobrushin domains should converge to SLE(3).
Of course, one should be careful about the way the graph is drawn in the plane.For instance, the isotropic spin Ising model of Section 2, when considered on a stretchedsquare lattice (every square is replaced by a rectangle), is not conformally invariant (it isnot invariant under rotations). Isoradial graphs form a large family of graphs possessinga natural embedding on which a critical Ising model is expected to be conformallyinvariant. More details are now provided about this fact.
Definition 8.1. A rhombic embedding of a graph G is a planar quadrangulationsatisfying the following properties:
the vertices of the quadrangulation are the vertices of G and Gâ, the edges connect vertices of G to vertices of Gâ corresponding to adjacent faces
of G, all the edges of the quadrangulation have equal length, see Fig. 10.
A graph which admits a rhombic embedding is called isoradial.
Isoradial graphs are fundamental for two reasons. First, discrete complex analy-sis on isoradial graphs was extensively studied (see e.g. [Mer01, Ken02, CS08]) asexplained in Section 4. Second, the Ising model on isoradial graphs satisfies very spe-cific integrability properties and a natural critical point can be defined as follows. LetJxy = arctanh[tan (Ξ/2)] where Ξ is the half-angle at the corner x (or equivalently y)made by the rhombus associated to the edge [xy]. One can define the critical Isingmodel with Hamiltonian
H(Ï) = ââxâŒy
JxyÏxÏy.
This Ising model on isoradial graphs (with rhombic embedding) is critical and confor-mally invariant in the following sense:
Theorem 8.2 (Chelkak, Smirnov [CS09]). The interfaces of the critical Ising modelon isoradial graphs converge, as the mesh size goes to 0, to the chordal Schramm-LoewnerEvolution with Îș = 3.
Note that the previous theorem is uniform on any rhombic graph discretizing a givendomain (Ω, a, b), as soon as the edge-length of rhombi is small enough. This provides afirst step towards universality for the Ising model.
Question 8.3. Since not every topological quadrangulation admits a rhombic embed-ding [KS05], can another embedding with a sufficiently nice version of discrete complexanalysis always be found?
Question 8.4. Is there a more general discrete setup where one can get similarestimates, in particular convergence of preholomorphic functions to the holomorphicones in the scaling limit?
In another direction, consider a biperiodic lattice L (one can think of the universalcover of a finite graph on the torus), and define a Hamiltonian with periodic correlations(Jxy) by setting H(Ï) = ââxâŒy JxyÏxÏy. The Ising model with this Hamiltonian makesperfect sense and there exists a critical inverse temperature separating the disorderedphase from the ordered phase.
Question 8.5. Prove that there always exists an embedding of L such that the Isingmodel on L is conformally invariant.
266 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
0 1
edge-weight p
pc(q) =âq
1+âq
cluster-weight q
subcritical phase
supercritical phase
critical phase: first order
critical phase: SLE(
4Ïarccos(â
âq/2)
)
1
2
4
percolation
FK Ising
4-colours Potts model representation
UST
Figure 17. The phase diagram of the FK percolation model on thesquare lattice.
8.2. Full scaling limit of critical Ising model. It has been proved in [KS10]that the scaling limit of Ising interfaces in Dobrushin domains is SLE(3). The nextquestion is to understand the full scaling limit of the interfaces. This question raisesinteresting technical problems. Consider the Ising model with free boundary conditions.Interfaces now form a family of loops. By consistency, each loop should look like aSLE(3). In [HK11], Hongler and KytolĂ€ made one step towards the complete pictureby studying interfaces with +/ â /free boundary conditions.
Sheffield and Werner [SW10a, SW10b] introduced a one-parameter family of pro-cesses of non-intersecting loops which are conformally invariant â called the ConformalLoop Ensembles CLE(Îș) for Îș > 8/3. Not surprisingly, loops of CLE(Îș) are locally simi-lar to SLE(Îș), and these processes are natural candidates for the scaling limits of planarmodels of statistical physics. In the case of the Ising model, the limits of interfaces alltogether should be a CLE(3).
8.3. FK percolation for general cluster-weight q â„ 0. The FK percolationwith cluster-weight q â (0,â) is conjectured to be critical for pc(q) = âq/(1 +âq) (see[BDC10] for the case q â„ 1). Critical FK percolation is expected to exhibit a very richphase transition, whose properties depend strongly on the value of q (see Fig. 17). Weuse generalizations of the FK fermionic observable to predict the critical behavior forgeneral q.
8.3.1. Case 0 †q †4. The critical FK percolation in Dobrushin domains can beassociated to a loop model exactly like the FK-Ising model: each loop receives a weightâq. In this context, one can define a natural generalization of the fermionic observable
on medial edges, called a parafermionic observable, by the formula
F (e) = EΩΎ,aÎŽ,bÎŽ,p,q[eÏâ iWÎł (e,bÎŽ)1eâÎł],(8.1)
CONFORMAL INVARIANCE OF LATTICE MODELS 267
where Ï = Ï(q) is called the spin (Ï takes a special value described below). Lemma 5.2has a natural generalization to any q â [0,â):
Proposition 8.6. For q †4 and any FK Dobrushin domain, consider the observableF at criticality with spin Ï = 1 â 2
Ïarccos(âq/2). For any medial vertex inside the
domain,
F (N) âF (S) = i[F (E) â F (W )](8.2)
where N , E, S and W are the four medial edges adjacent to the vertex.
These relations can be understood as Cauchy-Riemann equations around some ver-tices. Importantly, F is not determined by these relations for general q (the number ofvariables exceeds the number of equations). For q = 2, which corresponds to Ï = 1/2, thecomplex argument modulo Ï of the observable offers additional relations (Lemma 5.1)and it is then possible to obtain the preholomophicity (Proposition 5.3).
Parafermionic observables can be defined on medial vertices by the formula
F (v) = 1
2âeâŒv
F (e)where the summation is over medial edges with v as an endpoint. Even though they areonly weakly-holomorphic, one still expects them to converge to a holomorphic function.The natural candidate for the limit is not hard to find:
Conjecture 8.7. Let q †4 and (Ω, a, b) be a simply connected domain with twopoints on its boundary. For every z â Ω,
1(2ÎŽ)ÏFÎŽ(z) â ÏâČ(z)Ï when ÎŽ â 0(8.3)
where Ï = 1 â 2Ïarccos(âq/2), FÎŽ is the observable (at pc(q)) in discrete domains with
spin Ï, and Ï is any conformal map from Ω to R Ă (0,1) sending a to ââ and b to â.
Being mainly interested in the convergence of interfaces, one could try to follow thesame program as in Section 6:
Prove compactness of the interfaces. Show that sub-sequential limits are Loewner chains (with unknown random
driving process Wt). Prove the convergence of discrete observables (more precisely martingales) of
the model. Extract from the limit of these observables enough information to evaluate the
conditional expectation and quadratic variation of increments of Wt (in orderto harness the LĂ©vy theorem). This would imply that Wt is the Brownianmotion with a particular speed Îș and so curves converge to SLE(Îș).
The third step, corresponding to Conjecture 8.7, should be the most difficult. Notethat the first two steps are also open for q â 0,1,2. Even though the convergence ofobservables is still unproved, one can perform a computation similar to the proof ofProposition 6.7 in order to identify the possible limiting curves (this is the fourth step).The following conjecture is thus obtained:
Conjecture 8.8. For q †4, the law of critical FK interfaces converges to theSchramm-Loewner Evolution with parameter Îș = 4Ï/arccos(ââq/2).
The conjecture was proved by Lawler, Schramm and Werner [LSW04a] for q = 0,when they showed that the perimeter curve of the uniform spanning tree converges toSLE(8). Note that the loop representation with Dobrushin boundary conditions stillmakes sense for q = 0 (more precisely for the model obtained by letting q â 0 andp/q â 0). In fact, configurations have no loops, just a curve running from a to b (which
268 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
then necessarily passes through all the edges), with all configurations being equallyprobable. The q = 2 case corresponds to Theorem 3.13. All other cases are wide open.The q = 1 case is particularly interesting, since it is actually bond percolation on thesquare lattice.
8.3.2. Case q > 4. The picture is very different and no conformal invariance is ex-pected to hold. The phase transition is conjectured to be of first order : there aremultiple infinite-volume measures at criticality. In particular, the critical FK perco-lation with wired boundary conditions should possess an infinite cluster almost surelywhile the critical FK percolation with free boundary conditions should not (in this case,the connectivity probabilities should even decay exponentially fast). This result is knownonly for q â„ 25.72 (see [Gri06] and references therein).
The observable still makes sense in the q > 4 case, providing Ï is chosen so that2 sin(ÏÏ/2) = âq. Interestingly, Ï becomes purely imaginary in this case. A naturalquestion is to relate this change of behavior for Ï with the transition between conformallyinvariant critical behavior and first order critical behavior. Let us mention that theobservable was used to compute the critical point on isoradial graphs for q â„ 4 [BDS12]and to show that the phase transition is second order for 1 †q †4 [Dum12].
8.4. O(n) models on the hexagonal lattice. The Ising fermionic observablewas introduced in [Smi06] in the setting of general O(n) models on the hexagonallattice. This model, introduced in [DMNS81] on the hexagonal lattice, is a latticegas of non-intersecting loops. More precisely, consider configurations of non-intersectingsimple loops on a finite subgraph of the hexagonal lattice and introduce two parameters:a loop-weight n â„ 0 (in fact n â„ â2) and an edge-weight x > 0, and ask the probabilityof a configuration to be proportional to n# loopsx# edges.
Alternatively, an interface between two boundary points could be added: in thiscase configurations are composed of non-intersecting simple loops and one self-avoidinginterface (avoiding all the loops) from a to b.
The O(0)model is the self-avoiding walk, since no loop is allowed (there is still a self-avoiding path from a to b). The O(1) model is the high-temperature expansion of theIsing model on the hexagonal lattice. For integers n, the O(n)-model is an approximationof the high-temperature expansion of spin O(n)-models (models for which spins are n-dimensional unit vectors).
The physicist Bernard Nienhuis [Nie82, Nie84] conjectured that O(n)-modelsin the range n â (0,2) (after certain modifications n â (â2,2) would work) exhibit aBerezinsky-Kosterlitz-Thouless phase transition [Ber72, KT73]:
Conjecture 8.9. Let xc(n) = 1/â2 +â2 â n. For x < xc(n) (resp. x â„ xc(n)) the
probability that two points are on the same loop decays exponentially fast (as a powerlaw).
The conjecture was rigorously established for two cases only. When n = 1, thecritical value is related to the critical temperature of the Ising model. When n = 0,
it was recently proved in [DCS10] thatâ2 +â2 is the connective constant of the
hexagonal lattice.It turns out that the model exhibits one critical behavior at xc(n) and another on
the interval (xc(n),+â), corresponding to dilute and dense phases (when in the limitthe loops are simple and non-simple respectively), see Fig. 18. In addition to this, thetwo critical regimes are expected to be conformally invariant.
Exactly as in the case of FK percolation, the definition of the spin fermionic observ-able can be extended. For a discrete domain Ω with two points on the boundary a and
CONFORMAL INVARIANCE OF LATTICE MODELS 269
xc(n) =1
â
2+â
2ân
critical phase 2: SLE( 4Ïarccos(ân/2)
)
critical phase 1: SLE( 4Ï2Ïâarccos(ân/2)
)
subcritical phase
edge-weight x
loop-weight n
2
0 1/â
2 +â
2
1/â
2
1/â
31
Ising
XY model
Self-avoiding walk
Figure 18. The phase diagram of the O(n) model on the hexagonal lattice.
b, the parafermionic observable is defined on middle of edges by
F (z) = âÏâE(a,z) eâÏiWÎł(a,z)x# edges in Ïn# loops in Ï
âÏâE(a,b) eâÏiWÎł(a,b)x# edges in Ïn# loops in Ï(8.4)
where E(a, z) is the set of configurations of loops with one interface from a to z. Onecan easily prove that the observable satisfies local relations at the (conjectured) criticalvalue if Ï is chosen carefully.
Proposition 8.10. If x = xc(n) = 1/â2 +â2 â n, let F be the parafermionic ob-
servable with spin Ï = Ï(n) = 1 â 34Ï
arccos(ân/2); then
(p â v)F (p) + (q â v)F (q) + (r â v)F (r) = 0(8.5)
where p, q and r are the three mid-edges adjacent to a vertex v.
This relation can be seen as a discrete version of the Cauchy-Riemann equation onthe triangular lattice. Once again, the relations do not determine the observable forgeneral n. Nonetheless, if the family of observables is precompact, then the limit shouldbe holomorphic and it is natural to conjecture the following:
Conjecture 8.11. Let n â [0,2] and (Ω, a, b) be a simply connected domain withtwo points on the boundary. For x = xc(n),
FÎŽ(z)â (ÏâČ(z)ÏâČ(b))
Ï
(8.6)
where Ï = 1 â 34Ï
arccos(ân/2), FÎŽ is the observable in the discrete domain with spin Ï
and Ï is any conformal map from Ω to the upper half-plane sending a to â and b to 0.
A conjecture on the scaling limit for the interface from a to b in the O(n) modelcan also be deduced from these considerations:
Conjecture 8.12. For n â [0,2) and xc(n) = 1/â2 +â2 â n, as the mesh size goes
to zero, the law of O(n) interfaces converges to the chordal Schramm-Loewner Evolutionwith parameter Îș = 4Ï/(2Ï â arccos(ân/2)).
This conjecture is only proved in the case n = 1 (Theorem 2.10). The other casesare open. The case n = 0 is especially interesting since it corresponds to self-avoiding
270 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
walks. Proving the conjecture in this case would pave the way to the computation ofmany quantities, including the mean-square displacement exponent; see [LSW04b] forfurther details on this problem.
The phase x < xc(n) is subcritical and not conformally invariant (the interfaceconverges to the shortest curve between a and b for the Euclidean distance). The criticalphase x â (xc(n),â) should be conformally invariant, and universality is predicted:the interfaces are expected to converge to the same SLE. The edge-weight xc(n) =1/â2 â
â2 â n, which appears in Nienhuisâs works [Nie82, Nie84], seems to play a
specific role in this phase. Interestingly, it is possible to define a parafermionic observableat xc(n) with a spin Ï(n) other than Ï(n):
Proposition 8.13. If x = xc(n), let F be the parafermionic observable with spinÏ = Ï(n) = â 1
2â 3
4Ïarccos(ân/2); then
(p â v)F (p) + (q â v)F (q) + (r â v)F (r) = 0(8.7)
where p, q and r are the three mid-edges adjacent to a vertex v.
A convergence statement corresponding to Conjecture 8.11 for the observable withspin Ï enables to predict the value of Îș for xc(n), and thus for every x > xc(n) thanksto universality.
Conjecture 8.14. For n â [0,2) and x â (xc(n),â), as the lattice step goes tozero, the law of O(n) interfaces converges to the chordal Schramm-Loewner Evolutionwith parameter Îș = 4Ï/arccos(ân/2).
The case n = 1 corresponds to the subcritical high-temperature expansion of the Isingmodel on the hexagonal lattice, which also corresponds to the supercritical Ising modelon the triangular lattice via Kramers-Wannier duality. The interfaces should convergeto SLE(6). In the case n = 0, the scaling limit should be SLE(8), which is space-filling.For both cases, a (slightly different) model is known to converge to the correspondingSLE (site percolation on the triangular lattice for SLE(6), and the perimeter curve ofthe uniform spanning tree for SLE(8)). Yet, the known proofs do not extend to thiscontext. Proving that the whole critical phase (xc(n),â) has the same scaling limitwould be an important example of universality (not on the graph, but on the parameterthis time).
The two previous sections presented a program to prove convergence of discretecurves towards the Schramm-Loewner Evolution. It was based on discrete martingalesconverging to continuous SLE martingales. One can study directly SLE martingales (i.e.with respect to Ï(Îł[0, t])). In particular, gâČt(z)α[gt(z)âWt]ÎČ is a martingale for SLE(Îș)where Îș = 4(α â ÎČ)/[ÎČ(ÎČ â 1)]. All the limits in these notes are of the previous forms,see e.g. Proposition 6.7. Therefore, the parafermionic observables are discretizations ofvery simple SLE martingales.
Question 8.15. Can new preholomorphic observables be found by looking at dis-cretizations of more complicated SLE martingales?
Conversely, in [SS05], the harmonic explorer is constructed in such a way that anatural discretization of a SLE(4) martingale is a martingale of the discrete curve. Thisfact implied the convergence of the harmonic explorer to SLE(4).
Question 8.16. Can this reverse engineering be done for other values of Îș in orderto find discrete models converging to SLE?
8.5. Discrete observables in other models. The study can be generalized to avariety of lattice models, see the work of Cardy, Ikhlef, Riva, Rajabpour [IC09, RC07,RC06]. Unfortunately, the observable is only partially preholomorphic (satisfying only
CONFORMAL INVARIANCE OF LATTICE MODELS 271
Figure 19. Different possible plaquettes with their associated weights.
some of the Cauchy-Riemann equations) except for the Ising case. Interestingly, weightsfor which there exists a half-holomorphic observable which is not degenerate in thescaling limit always correspond to weights for which the famous Yang-Baxter equalityholds.
Question 8.17. The approach to two-dimensional integrable models described hereis in several aspects similar to the older approaches based on the Yang-Baxter relations[Bax89]. Can one find a direct link between the two approaches?
Let us give the example of the O(n) model on the square lattice. We refer to [IC09]for a complete study of the following.
It is tempting to extend the definition ofO(n)models to the square lattice in order toobtain a family of models containing self-avoiding walks on Z
2 and the high-temperatureexpansion of the Ising model. Nevertheless, difficulties arise when dealing with O(n)models on non-trivalent graphs. Indeed, the indeterminacy when counting intersectingloops prevents us from defining the model as in the previous subsection.
One can still define a model of loops on G â L by distinguishing between localconfigurations: faces of Gâ â Lâ are filled with one of the nine plaquettes in Fig. 19. Aweight pv is associated to every face v â Gâ depending on the type of the face (meaning itsplaquette). The probability of a configuration is then proportional to n# loopsâvâLâ pv.
Remark 8.18. The case u1 = u2 = v = x, t = 1 and w1 = w2 = n = 0 corresponds tovertex self-avoiding walks on the square lattice. The case u1 = u2 = v =
âw1 =
âw2 = x
and n = t = 1 corresponds to the high-temperature expansion of the Ising model. Thecase t = u1 = u2 = v = 0, w1 = w2 = 1 and n > 0 corresponds to the FK percolation atcriticality with q = n.
A parafermionic observable can also be defined on the medial lattice:
F (z) = âÏâE(a,z) eâiÏWÎł(a,z) n# loops âvâLâ pvâÏâE n# loops âvâLâ pv(8.8)
where E corresponds to all the configurations of loops on the graph, and E(a, z) corre-sponds to configurations with loops and one interface from a to z.
One can then look for a local relation for F around a vertex v, which would be adiscrete analogue of the Cauchy-Riemann equation:
(8.9) F (N)â F (S) = i[F (E) âF (W )],An additional geometric degree of freedom can be added: the lattice can be stretched,meaning that each rhombus is not a square anymore, but a rhombus with inside angleα.
As in the case of FK percolations and spin Ising, one can associate configurationsby pairs, and try to check (8.9) for each of these pairs, thus leading to a certain number
272 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
of complex equations. We possess degrees of freedom in the choice of the weights of themodel, of the spin Ï and of the geometric parameter α. Very generally, one can thus tryto solve the linear system and look for solutions. This leads to the following discussion:
Case v = 0 and n = 1: There exists a non-trivial solution for every spin Ï, whichis in bijection with a so-called six-vertex model in the disordered phase. The heightfunction associated with this model should converge to the Gaussian free field. Thisis an example of a model for which interfaces cannot converge to SLE (in [IC09]; it isconjectured that the limit is described by SLE(4, Ï)).
Case v = 0 and n â 1: There exist unique weights associated to an observable withspin â1. This solution is in bijection with the FK percolation at criticality with
âq =
n+ 1. Nevertheless, physical arguments tend to show that the observable with this spinshould have a trivial scaling limit. It would not provide any information on the scalinglimit of the model itself; see [IC09] for additional details.
Case v â 0: Fix n. There exists a solution for Ï = 3η
2Ïâ 1
2where η â [âÏ,Ï] satisfies
ân2= cos2η. Note that there are a priori four possible choices for Ï. In general the
following weights can be found:â§âȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâšâȘâȘâȘâȘâȘâȘâȘâȘâȘâȘâ©
t = â sin(2Ï â 3η/2) + sin(5η/2) â sin(3η/2) + sin(η/2)u1 = â2 sin(η) cos(3η/2 â Ï)u2 = â2 sin(η) sin(Ï)v = â2 sin(Ï) cos(3η/2 â Ï)w1 = â2 sin(Ï â η) cos(3η/2 â Ï)w2 = 2 cos(η/2 â Ï) sin(Ï)
where Ï = (1 + Ï)α. We now interpret these results:When η â [0, Ï], the scaling limit has been argued to be described by a Coulomb gas
with a coupling constant 2η/Ï. In other words, the scaling limit should be the same asthe corresponding O(n) model on the hexagonal lattice. In particular, interfaces shouldconverge to the corresponding Schramm-Loewner Evolution.
When η â [âÏ,0], the scaling limit curve cannot be described by SLE, and it pro-vides yet another example of a two-dimensional model for which the scaling limit is notdescribed via SLE.
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276 HUGO DUMINIL-COPIN AND STANISLAV SMIRNOV
Département de Mathématiques, Université de GenÚve, GenÚve, Switzerland
E-mail address: [email protected]
Département de Mathématiques, Université de GenÚve, GenÚve, Switzerland. Cheby-
shev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg,
199178 Russia
E-mail address: [email protected]