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



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.





Question 1

for which T does 〈σ(x)σ(y)〉 = E[σ(x)σ(y)] vanish when y goesto infinity (zero magnetization)?





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?




massive random walk representation of correlationsLoop representation of the Ising modelFrom the 2D-Ising model to the loop representation


(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





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)





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).




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

].





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).





Putting all the pieces together, we obtain

〈σ(a)σ(b)〉 � Gm(a, b).


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.





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).


zero-magnetization at criticality

power law decay for correlations

convergence of the interface to SLE(16/3) for the randomcluster model





Thank you for your attention


2D-Ising model and random walks - Duke Universityrtd/CPSS2009/HDC.pdf · 2D-Ising model and random...

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