Ornstein-Zernike theory and some applications - Introductory lecture · 2018. 6. 28. · I Ornstein–Zernike 1914, Zernike 1916: jAj= jBj= 1 non-rigorous I Abraham–Kunz 1977, Paes-Leme

Ornstein-Zernike theory and some applications

Introductory lecture

Yvan Velenik

Université de Genève

These slides: www.unige.ch/math/folks/velenik/Ischia/IschiaLec1.pdfNotes for the next few hours: www.unige.ch/math/folks/velenik/Ischia/IschiaLN.pdf

Universality in Statistical Mechanics

In statistical mechanics, universality is usually associated to the behavior of systemsat a critical point. Among its many features:

I Many quantities exhibiting power-law behavior with universal exponents, sharedby all systems in large universality classes.

I These exponents often exhibit nontrivial dependence on the spatial dimension d.I Existence of an upper critical dimension du:

I When d > du, the dependence on d has a “simple” functional form (mean �eld).I When d = du, often presence of logarithmic corrections.I When d < du, the dependence on the dimension does not display a simple pattern.

However, universal behavior can also be observed away from the critical point andcan display a similar phenomenology. We are now going to discuss some examples inthe Ising and Potts models.

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Universality in Statistical Mechanics

In statistical mechanics, universality is usually associated to the behavior of systemsat a critical point. Among its many features:

I Many quantities exhibiting power-law behavior with universal exponents, sharedby all systems in large universality classes.

I These exponents often exhibit nontrivial dependence on the spatial dimension d.I Existence of an upper critical dimension du:

I When d > du, the dependence on d has a “simple” functional form (mean �eld).I When d = du, often presence of logarithmic corrections.I When d < du, the dependence on the dimension does not display a simple pattern.

However, universal behavior can also be observed away from the critical point andcan display a similar phenomenology. We are now going to discuss some examples inthe Ising and Potts models.

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Part 1

Asymptotics of correlations in the Ising model

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Ising model on Zd

I Con�gurations: Let B(N) = {−N, . . . ,N}d.

ΩN ={σ = (σi)i∈B(N) : σi ∈ {−1, 1}

}

I Energy of σ ∈ ΩN:

H(σ) = −∑{i,j}⊂BN

i∼j

σiσj

I Gibbs measure in B(N) at inverse temperature β ≥ 0:

µN;β(σ) =1ZN;β

e−βH(σ)

I In�nite-volume Gibbs measure:

µβ = limN→∞

µN;β

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Ising model on Zd

ΩN ={σ = (σi)i∈B(N) : σi ∈ {−1, 1}

}

H(σ) = −∑{i,j}⊂BN

i∼j

σiσj

µN;β(σ) =1ZN;β

e−βH(σ)

µβ = limN→∞

µN;β

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Ising model on Zd

ΩN ={σ = (σi)i∈B(N) : σi ∈ {−1, 1}

}

H(σ) = −∑{i,j}⊂BN

i∼j

σiσj

µN;β(σ) =1ZN;β

e−βH(σ)

µβ = limN→∞

µN;β

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Ising model on Zd

ΩN ={σ = (σi)i∈B(N) : σi ∈ {−1, 1}

}

H(σ) = −∑{i,j}⊂BN

i∼j

σiσj

µN;β(σ) =1ZN;β

e−βH(σ)

µβ = limN→∞

µN;β

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Ising model on Zd

For any d ≥ 2, there exists βc = βc(d) ∈ (0,∞) such that

∀β < βc There is no long-range order:

lim‖i‖→∞

µβ(σ0 = σi) =12

∀β > βc There is long-range order:

lim‖i‖→∞

µβ(σ0 = σi) >12

We assume that d ≥ 2 and β < βc(d) . We denote by 〈·〉β the expectation w.r.t. µβ .

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Ising model on Zd

lim‖i‖→∞

µβ(σ0 = σi) =12

lim‖i‖→∞

µβ(σ0 = σi) >12

We assume that d ≥ 2 and β < βc(d) . We denote by 〈·〉β the expectation w.r.t. µβ .

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Decay of correlations

As mentioned in the previous slide, when β < βc,

lim‖i‖→∞

〈σ0σi〉β = lim‖i‖→∞

2µβ(σ0 = σi)− 1 = 0.

Question: how fast is this decay?

Let [x] ∈ Zd be the coordinate-wise integer part of x ∈ Rd.

Theorem [Aizenman, Barsky, Fernández 1987]

For all β < βc(d) and any unit-vector u in Rd, the inverse correlation length

ξβ(u) = − limn→∞

1n

log〈σ0σ[nu]〉β

exists and is positive.

What about sharper asymptotics and more general functions?

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lim‖i‖→∞

2µβ(σ0 = σi)− 1 = 0.

1n

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lim‖i‖→∞

2µβ(σ0 = σi)− 1 = 0.

1n

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Asymptotics of correlations

Let f , g be two local functions and denote by θx the translation by x ∈ Zd.Let Covβ(f , g) = 〈fg〉β − 〈f〉β〈g〉β .

What is the asymptotic behavior of

Covβ(f , θ[nu]g)as n→∞?

Let σA =∏

i∈A σi. Writing

f =∑

A⊂supp(f)

f̂AσA, g =∑

B⊂supp(g)

ĝBσB,

yieldsCovβ(f , θ[nu]g) =

∑A⊂supp(f)B⊂supp(g)

f̂AĝB Covβ(σA, σB+[nu]).

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Asymptotics of correlations

Let f , g be two local functions and denote by θx the translation by x ∈ Zd.Let Covβ(f , g) = 〈fg〉β − 〈f〉β〈g〉β .

What is the asymptotic behavior of

Covβ(f , θ[nu]g)as n→∞?

Let σA =∏

i∈A σi. Writing

f =∑

A⊂supp(f)

f̂AσA, g =∑

B⊂supp(g)

ĝBσB,

yieldsCovβ(f , θ[nu]g) =

∑A⊂supp(f)B⊂supp(g)

f̂AĝB Covβ(σA, σB+[nu]).

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This motivates the following

Main question

What is the asymptotic behavior, as n→∞, of

Covβ(σA, σB+[nu])

for A, B b Zd and u a unit-vector in Rd?

Of course, by symmetry, 〈σC〉β = 0 whenever |C| is odd.

Covβ(σA, σB) = 0 whenever |A|+ |B| is odd.

We are thus left with two cases to consider:

Odd-odd correlations

|A|, |B| both oddEven-even correlations

|A|, |B| both even

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Main question

|A|, |B| both even

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Main question

|A|, |B| both even

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Best result to date:

Theorem [Campanino, Io�e, V. 2004]

Let d ≥ 2 and β < βc(d). Let A, B b Zd with |A| and |B| odd.For any unit-vector u, there exists a constant 0 < C < ∞ (depending onA, B, u, β) such that

Covβ(σA, σB+[nu]) =C

n(d−1)/2e−ξβ (u)n (1 + o(1)),

as n→∞.

This result has a long history. Some milestones:

I Ornstein–Zernike 1914, Zernike 1916: |A| = |B| = 1 non-rigorousI Abraham–Kunz 1977, Paes-Leme 1978: |A| = |B| = 1 β � 1I Bricmont–Fröhlich 1985, Minlos-Zhizhina 1988, 1996: |A|, |B| odd β � 1I Campanino–Io�e–V. 2003: |A| = |B| = 1 β < βc(d)

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Best result to date:

Theorem [Campanino, Io�e, V. 2004]

Let d ≥ 2 and β < βc(d). Let A, B b Zd with |A| and |B| odd.For any unit-vector u, there exists a constant 0 < C < ∞ (depending onA, B, u, β) such that

Covβ(σA, σB+[nu]) =C

n(d−1)/2e−ξβ (u)n (1 + o(1)),

as n→∞.

This result has a long history. Some milestones:

I Ornstein–Zernike 1914, Zernike 1916: |A| = |B| = 1 non-rigorousI Abraham–Kunz 1977, Paes-Leme 1978: |A| = |B| = 1 β � 1I Bricmont–Fröhlich 1985, Minlos-Zhizhina 1988, 1996: |A|, |B| odd β � 1I Campanino–Io�e–V. 2003: |A| = |B| = 1 β < βc(d)

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Even-even correlations

Substantially more delicate!

The analysis started with the case |A| = |B| = 2. Physicists quickly understood that

Covβ(σA, σB+[nu]) = e−2ξβ (u)n (1+o(1)).

However, concerning the prefactor, two con�icting predictions were put forward:

Polyakov 1969 Camp, Fisher 1971n−2 d = 2(n log n)−2 d = 3n−(d−1) d ≥ 4

n−d for all d ≥ 2

(Note that both coincided with the (already known) exact computation when d = 2.)

It turns out that Polyakov was right. This was �rst shown in

I Bricmont–Fröhlich 1985: |A| = |B| = 2 β � 1 d ≥ 4I Minlos–Zhizhina 1988, 1996: |A|, |B| even β � 1 d ≥ 2

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Let Ξ(n) =

n2 when d = 2,

(n log n)2 when d = 3,

nd−1 when d ≥ 4.

Theorem [Ott, V. 2018]

Let d ≥ 2 and β < βc(d). Let A, B b Zd with |A| and |B| even.For any unit vector u inRd, there exist constants 0 < C− ≤ C+

Let Ξ(n) =

n2 when d = 2,

(n log n)2 when d = 3,

nd−1 when d ≥ 4.

Theorem [Ott, V. 2018]

Let d ≥ 2 and β < βc(d). Let A, B b Zd with |A| and |B| even.For any unit vector u inRd, there exist constants 0 < C− ≤ C+

Heuristics (and di�culties)

It is well-known that Ising spin correlations can be expressed in terms of disjointpaths using the high-temperature representation: for any C = {x1, . . . , x2n} ⊂ Zd,

〈σC〉β =∑

pairings

∑γ1:xi1→xj1

· · ·∑

γn:xin→xjn

w(γ1, . . . , γn)

for some (complicated) weight w.

x2

x5

γ3

x1

γ1x3

x4 x6γ2

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Let A = {x, y} and B + [nu] = {u, v}. High-temperature expansion of 〈σxσyσuσv〉βyields 3 types of con�gurations:

γ2xy

γ1uv

uv

xy

uv

xy

High-temperature expansion of 〈σxσy〉β〈σuσv〉β yields

γ2uv

xy

γ1

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Let A = {x, y} and B + [nu] = {u, v}. High-temperature expansion of 〈σxσyσuσv〉βyields 3 types of con�gurations:

γ2xy

γ1uv

uv

xy

uv

xy

High-temperature expansion of 〈σxσy〉β〈σuσv〉β yields

γ2uv

xy

γ1

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Now, since β < βc(d), one may expect the paths γ1 and γ2 to stay far from each other,so that the expectation factorizes and

γ2xy

γ1uv ≈ γ2 uv

xy

γ1

Then, Covβ(σA, σB+[nu]) would be dominated by the contributions of

uv

xy

anduv

xy

Then, neglecting the interactions between γ1, γ2 would yield

Covβ(σA, σB+[nu]) ≈ 〈σxσu〉β〈σyσv〉β + 〈σxσv〉β〈σyσu〉β ≈ n−(d−1)e−2ξβ (u)n,

which is what we want when d ≥ 4.

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γ2xy

γ1uv ≈ γ2 uv

xy

γ1

uv

xy

anduv

xy

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γ2xy

γ1uv ≈ γ2 uv

xy

γ1

uv

xy

anduv

xy

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Finally, assuming that the paths behave as a pair of non-intersecting (massive)random walk bridges would then yield an additional factor of order

n−1 when d = 2,

(log n)−2 when d = 3,

1 when d ≥ 4,

which would lead to the desired behavior...

Unfortunately, such an approach su�ers from several problems...

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Finally, assuming that the paths behave as a pair of non-intersecting (massive)random walk bridges would then yield an additional factor of order

n−1 when d = 2,

(log n)−2 when d = 3,

1 when d ≥ 4,

which would lead to the desired behavior...

Unfortunately, such an approach su�ers from several problems...

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Problems with this argument

I In fact,

γ2xy

γ1uv − γ2 uv

xy

γ1= e−2ξβ (u)n (1+o(1))

and thus cannot be neglected!

I The paths γ1, γ2 have long-distance (self)interactions.In particular, w(γ1, γ2) 6= w(γ1)w(γ2).

I Even if we could factor the weights, it is not at all obvious why thenon-intersection constraint should yield the same behavior as if γ1 and γ2 wererandom walk bridges.

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I In fact,

γ2xy

γ1uv − γ2 uv

xy

γ1= e−2ξβ (u)n (1+o(1))

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I In fact,

γ2xy

γ1uv − γ2 uv

xy

γ1= e−2ξβ (u)n (1+o(1))

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To solve the �rst two problems, we use

I various graphical representations, in order to reduce to two independent objects(HT paths or FK clusters), conditioned on not intersecting:

To solve the third one, we rely on

I the Ornstein-Zernike theory (Campanino–Io�e–V. 2003, 2008 and Ott–V. 2017), inorder to approximate these objects using directed random walks on Zd:

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To solve the �rst two problems, we use

I various graphical representations, in order to reduce to two independent objects(HT paths or FK clusters), conditioned on not intersecting:

To solve the third one, we rely on

I the Ornstein-Zernike theory (Campanino–Io�e–V. 2003, 2008 and Ott–V. 2017), inorder to approximate these objects using directed random walks on Zd:

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Part 2

Potts model with a defect line

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Potts model

ΩN ={σ = (σi)i∈B(N) : σi ∈ {1, . . . , q}

}(The Ising model corresponds to q = 2)

H(σ) =∑{i,j}⊂BN

i∼j

1{σi 6=σj}

µN;β(σ) =1ZN;β

e−βH(σ)

µβ = limN→∞

µN;β

18/35

Potts model

ΩN ={σ = (σi)i∈B(N) : σi ∈ {1, . . . , q}

i∼j

1{σi 6=σj}

µN;β(σ) =1ZN;β

e−βH(σ)

µβ = limN→∞

µN;β

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Potts model

ΩN ={σ = (σi)i∈B(N) : σi ∈ {1, . . . , q}

i∼j

1{σi 6=σj}

µN;β(σ) =1ZN;β

e−βH(σ)

µβ = limN→∞

µN;β

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Potts model

ΩN ={σ = (σi)i∈B(N) : σi ∈ {1, . . . , q}

i∼j

1{σi 6=σj}

µN;β(σ) =1ZN;β

e−βH(σ)

µβ = limN→∞

µN;β

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Potts models

lim‖i‖→∞

µβ(σ0 = σi) =1q

lim‖i‖→∞

µβ(σ0 = σi) >1q

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Decay of spin-spin correlations at high temperatures

Fix β < βc .

Exponential decay of correlations

Theorem (Duminil-Copin, Raou�, Tassion – arXiv:1705.03104)

Let u be a unit vector in Rd. Then:

1n

log

∣∣∣µβ(σ0 = σ[nu])− 1q ∣∣∣

> 0

ξβ(u) is the inverse correlation length (in direction u)

Ornstein–Zernike asymptotics

Theorem (Campanino, Io�e, V. 2003, 2008)

There exists Ψβ : Sd−1 → (0,∞) such that, as n→∞,

µβ(σ0 = σ[nu]) =1q

+Ψβ(u)n(d−1)/2

e−ξβ (u)n (1 + o(1))

Moreover, Ψβ(·) and ξβ(·) depend analytically on the direction.

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Fix β < βc .

1n

log

∣∣∣µβ(σ0 = σ[nu])− 1q ∣∣∣

> 0

µβ(σ0 = σ[nu]) =1q

+Ψβ(u)n(d−1)/2

e−ξβ (u)n (1 + o(1))

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Fix β < βc .

1n

log∣∣∣µβ(σ0 = σ[nu])− 1q ∣∣∣ > 0

µβ(σ0 = σ[nu]) =1q

+Ψβ(u)n(d−1)/2

e−ξβ (u)n (1 + o(1))

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Fix β < βc .

1n

log∣∣∣µβ(σ0 = σ[nu])− 1q ∣∣∣ > 0

µβ(σ0 = σ[nu]) =1q

+Ψβ(u)n(d−1)/2

e−ξβ (u)n (1 + o(1))

Moreover, Ψβ(·) and ξβ(·) depend analytically on the direction.20/35

Introducing the defect line

I Defect line: L = {ke1 ∈ Zd : k ∈ Z}

I Coupling constants:

Jij =

1 if i ∼ j, {i, j} 6⊂ LJ if i ∼ j, {i, j} ⊂ L0 otherwise

HN;J(σ) =∑

{i,j}⊂B(N)

Jij 1{σi 6=σj}

I Gibbs measures µN;β,J and µβ,J: as before

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Two simulations

J = 1 J = 3

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Longitudinal correlation length

We assume that d ≥ 2 , β < βc and J ≥ 0 .

Longitudinal inverse correlation length:

ξβ(J) = − limn→∞

1n

log∣∣∣µβ,J(σ0 = σne1)− 1q ∣∣∣

Main questionHow does ξβ(J) vary as J grows from 0 to +∞?

This analysis had only been achieved (by McCoy and Perk in 1980) for the Ising modelon Z2Z2Z2, on the basis of exact computations.

Goal: extend (part of) their analysis to arbitrary Potts models and arbitrarydimensions. Note that both extensions destroy integrability.

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1n

log∣∣∣µβ,J(σ0 = σne1)− 1q ∣∣∣

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1n

log∣∣∣µβ,J(σ0 = σne1)− 1q ∣∣∣

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1n

log∣∣∣µβ,J(σ0 = σne1)− 1q ∣∣∣

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Basic properties

Theorem (Ott, V. – arXiv:1706.09130)

I J 7→ ξβ(J) is positive, Lipschitz-continuous and nonincreasing

I ξβ(J) = ξβ(1) for all J ≤ 1

I ξβ(J) ∼ e−βJ as J→∞

In particular, there exists Jc ∈ [1,∞) such that

ξβ(J) = ξβ(1) for all J ≤ Jc and ξβ(J) < ξβ(1) for all J > Jc

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Basic properties

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Basic properties

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Basic properties

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Basic properties — typical graph

JJc

ξβ(J)

ξβ(1)

(actually corresponds to the exact result by McCoy and Perk for the Ising model on Z2)

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Basic properties — typical graph

JJc

ξβ(J)

ξβ(1)

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Re�ned properties

?

JJc

ξβ(J)

ξβ(1)

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Re�ned properties

1. Jc = 1 when d = 2 or 3, but Jc > 1 when d ≥ 4

2. J 7→ ξβ(J) is decreasing and real-analytic when J > Jc3. There exist constants c±2 , c

±3 > 0 such that, as J ↓ Jc,

c−2 (J− Jc)2 ≤ ξβ(Jc)− ξβ(J) ≤ c+2 (J− Jc)2 (d = 2)

e−c−3 /(J−Jc) ≤ ξβ(Jc)− ξβ(J) ≤ e−c

+3 /(J−Jc) (d = 3)

4. For all J > Jc, there exists C = C(J, β) such that

µβ,J(σ0 = σne1) =1q

+ C e−ξβ (J)n (1 + o(1))

Predictions based on e�ective models yield the following conjecture when J < Jc:

µβ,J(σ0 = σne1)−1q∼

n−3/2 e−ξβ (J)n (d = 2)

n−1(log n)−2 e−ξβ (J)n (d = 3)

n−(d−1)/2 e−ξβ (J)n (d ≥ 4)

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Re�ned properties

+3 /(J−Jc) (d = 3)

+ C e−ξβ (J)n (1 + o(1))

n−3/2 e−ξβ (J)n (d = 2)

n−1(log n)−2 e−ξβ (J)n (d = 3)

n−(d−1)/2 e−ξβ (J)n (d ≥ 4)

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Re�ned properties

+3 /(J−Jc) (d = 3)

+ C e−ξβ (J)n (1 + o(1))

n−3/2 e−ξβ (J)n (d = 2)

n−1(log n)−2 e−ξβ (J)n (d = 3)

n−(d−1)/2 e−ξβ (J)n (d ≥ 4)

27/35

Re�ned properties

+3 /(J−Jc) (d = 3)

+ C e−ξβ (J)n (1 + o(1))

Predictions based on e�ective models yield the following conjecture:

ξβ(Jc)− ξβ(J) ∼

(J− Jc)2 (d = 4)(J− Jc)/| log(J− Jc)| (d = 5)J− Jc (d ≥ 6)

n−3/2 e−ξβ (J)n (d = 2)

n−1(log n)−2 e−ξβ (J)n (d = 3)

n−(d−1)/2 e−ξβ (J)n (d ≥ 4)

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Re�ned properties

+3 /(J−Jc) (d = 3)

+ C e−ξβ (J)n (1 + o(1))

n−3/2 e−ξβ (J)n (d = 2)

n−1(log n)−2 e−ξβ (J)n (d = 3)

n−(d−1)/2 e−ξβ (J)n (d ≥ 4)

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Re�ned properties

+3 /(J−Jc) (d = 3)

+ C e−ξβ (J)n (1 + o(1))

n−3/2 e−ξβ (J)n (d = 2)

n−1(log n)−2 e−ξβ (J)n (d = 3)

n−(d−1)/2 e−ξβ (J)n (d ≥ 4)

27/35

Interface localization

In dimension 2, the above analysis has also immediate implications for the behaviorof the Potts model at β > βc .

Namely, what is the e�ect that a row of modi�ed coupling constants on the statisticalproperties of an interface?

This problem was �rst considered in [Abraham JPA ’81] for the 2d Ising model, on the basisof exact computations.

Goal: extend (part of) his analysis to arbitrary 2d Potts models.

28/35

In dimension 2, the above analysis has also immediate implications for the behaviorof the Potts model at β > βc .

Namely, what is the e�ect that a row of modi�ed coupling constants on the statisticalproperties of an interface?

This problem was �rst considered in [Abraham JPA ’81] for the 2d Ising model, on the basisof exact computations.

Goal: extend (part of) his analysis to arbitrary 2d Potts models.

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2

1

2

1

n

All coupling constants are equal to 1 except those drawn in purple, whose the value isJ ≥ 0.

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Theorem (Campanino, Io�e, V. 2008)J = 1: the interface converges to a Brownian bridge under di�usive scaling.

Theorem (Ott, V. preprint ’17)J < 1: the interface is localized (converges to a line segment under di�usive scaling).

J = 1

J = 12

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Theorem (Campanino, Io�e, V. 2008)J = 1: the interface converges to a Brownian bridge under di�usive scaling.

Theorem (Ott, V. preprint ’17)J < 1: the interface is localized (converges to a line segment under di�usive scaling).

J = 1 J = 1230/35

Some rough ideas of the approach used

I Using the FK representation, the probability µβ,J(σ0 = σne1) can be expressed interms of a sum over clusters containg 0 and ne1, with some complicated weight.

I This may be somewhat reminiscent of the pinning problem for a(d− 1)-dimensional random walk.

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Some rough ideas of the approach used

I Closer analogy using Ornstein–Zernike theory [Campanino–Io�e–V. 2008, Ott–V. 2017].coupling between FK cluster and a directed random walk on Zd.

I With some work, one then obtains a variant of the RW pinning problem for adirected random walk interacting (in a non-trivial way) with the line L.

I One can then adapt some of the methods used to study the RW pinning problem.32/35

References i

M. Aizenman, D. J. Barsky, and R. Fernández, The phase transition in a generalclass of Ising-type models is sharp, J. Statist. Phys. 47 (1987), no. 3-4, 343–374.

D. B. Abraham, Binding of a domain wall in the planar Ising ferromagnet, J. Phys.A 14 (1981), no. 9, L369–L372.

D. B. Abraham and H. Kunz, Ornstein-Zernike theory of classical �uids at lowdensity, Phys. Rev. Lett. 39 (1977), no. 16, 1011–1014.

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Ornstein-Zernike theory and some applications - Introductory lecture · 2018. 6. 28. · I Ornstein–Zernike 1914, Zernike 1916: jAj= jBj= 1 non-rigorous I Abraham–Kunz 1977, Paes-Leme