2D-Ising model and random walks - Duke Universityrtd/CPSS2009/HDC.pdf · 2D-Ising model and random...
Transcript of 2D-Ising model and random walks - Duke Universityrtd/CPSS2009/HDC.pdf · 2D-Ising model and random...
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
2D-Ising model and random walks
Hugo Duminil-Copin, Universite de Geneve
july 2009
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
The 2D-Ising model on the square latticeAnswer to question 1
We consider the square lattice (say restricted to a finite graph D).
D
Any site x ∈ D has a (random) spin σ(x) that can be either -1 or+1.
The probability of a configuration σ is proportional to:
exp(− 1
TH(σ))
where − T is a temperature parameter
− H(σ) =∑i∼j
[1− σ(i)σ(j)] = #disagreements.
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
The 2D-Ising model on the square latticeAnswer to question 1
Question 1
for which T does 〈σ(x)σ(y)〉 = E[σ(x)σ(y)] vanish when y goesto infinity (zero magnetization)?
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
The 2D-Ising model on the square latticeAnswer to question 1
Answer to question 1
∞0subcriticalsupercritical critical
spontaneous magnetization no magnetization
Questions 2 and 3
For high temperature, can we identify the rate of decay?
At criticality, are we able to describe the model?
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
massive random walk representation of correlationsLoop representation of the Ising modelFrom the 2D-Ising model to the loop representation
Answer to question 2
(Onsager, Kaufman, Mc Coy...) The critical temperature isTc = 2/ log(1 +
√2) and there is exponential decay of correlations
above Tc .
(Beffara, D-C, Smirnov, 2009) For any T > Tc , there exists mT
such that〈σ(x)σ(y)〉 � GmT
(x , y)
for any sites x and y .
Let 0 ≤ m ≤ 1 and x , y ∈ Z2, the massive Green function of mass m isgiven by
Gm(x , y) = Ey [(1−m)τx ]
If 0 < m, then Gm(x , .) decays exponentially fast and we have very
good estimates on the behavior at infinity
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
massive random walk representation of correlationsLoop representation of the Ising modelFrom the 2D-Ising model to the loop representation
consider a finite grid in a domain D and the so-called randomcluster model on it:
type 1: open
type 2: closed
every edge is open (blue) or closed (red edge in the duallattice)
the probability of a configuration is given by:
x#open edges√q#loops
Z (x , q)
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
massive random walk representation of correlationsLoop representation of the Ising modelFrom the 2D-Ising model to the loop representation
Theorem
The 2D-Ising model at temperature T can be coupled with theloop model with parameters q = 2 and x = (e1/T − 1)/
√2).
/
,
Corollary
For any x , y ∈ Z2, 〈σ(x)σ(y)〉 = Px(x ↔ y).
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
Winding and fermionic observableMassive harmonicity of FAt criticality: Russo-Seymour-Welsh Theorem
We must transform the observable, introducing a weight term toevery configuration:
Definition
The winding W (a, b) of the curve is the ’number of clockwiseturns’ (in radian) between a and b.
a
b
a
b
We can define the so-called fermionic observable F :
Fx(a, b) = Ex
[e iW (a,b)/2
1a,b belong to the same loop
].
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
Winding and fermionic observableMassive harmonicity of FAt criticality: Russo-Seymour-Welsh Theorem
Using local bijections, one can show the following:
Lemma
For any x > 0, there exists a mass m = m(x) > 0 such that theobservable F (a, .) satisfies:
∆F (a, z) = −mF (a, z)
for every z 6= a. Moreover, m = m(x) = 0 iff x = 1.
The winding could drastically decrease the order of magnitudeof F . Nevertheless, for x 6= 1, one obtains that the winding istight so that:
Lemma
for any x < 1 and a, b, F (a, b) � P(a ↔ b).
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
Winding and fermionic observableMassive harmonicity of FAt criticality: Russo-Seymour-Welsh Theorem
Putting all the pieces together, we obtain
〈σ(a)σ(b)〉 � Gm(a, b).
Answer to question 2
there is exponential decay of the two points function forT > Tc
the critical point Tc is equal to 2/ log(1 +√
2)
the Ornstein-Zernike Theory holds:
〈σ(0)σ(nx)〉 ∼ c(x)n1/2e−ξ(x)n
when n goes to infinity.
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
Winding and fermionic observableMassive harmonicity of FAt criticality: Russo-Seymour-Welsh Theorem
At criticality, the observable becomes harmonic.
Theorem (D-C, Hongler, Nolin, 2009)
Then for any β > 0, there exists cβ , dβ such that:
cβ ≤ Pxc (βn
n
) ≤ dβ
for any rectangle with dimensions (βn, n).
Answer to question 3
zero-magnetization at criticality
power law decay for correlations
convergence of the interface to SLE(16/3) for the randomcluster model
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks
Probabilistic ModelAnswer to question 2: fermionic observable
Random walk study of the Ising model
Winding and fermionic observableMassive harmonicity of FAt criticality: Russo-Seymour-Welsh Theorem
Thank you for your attention
Hugo Duminil-Copin, Universite de Geneve 2D-Ising model and random walks