ornstein-zernike theory and some applications - introductory lecture · 2018. 6. 28. · i...
TRANSCRIPT
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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
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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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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
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
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
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
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
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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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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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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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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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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Decay of correlations
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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Decay of correlations
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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Decay of correlations
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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Odd-odd correlations
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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Odd-odd correlations
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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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
9/35
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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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Even-even correlations
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+
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Even-even correlations
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+
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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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Heuristics (and di�culties)
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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Heuristics (and di�culties)
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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Heuristics (and di�culties)
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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Heuristics (and di�culties)
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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Heuristics (and di�culties)
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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Heuristics (and di�culties)
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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Heuristics (and di�culties)
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.
15/35
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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.
15/35
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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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Problems with this argument
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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Problems with this argument
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
I Con�gurations: Let B(N) = {−N, . . . ,N}d.
ΩN ={σ = (σi)i∈B(N) : σi ∈ {1, . . . , q}
}(The Ising model corresponds to q = 2)
I Energy of σ ∈ ΩN:
H(σ) =∑{i,j}⊂BN
i∼j
1{σi 6=σj}
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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Potts model
I Con�gurations: Let B(N) = {−N, . . . ,N}d.
ΩN ={σ = (σi)i∈B(N) : σi ∈ {1, . . . , q}
}(The Ising model corresponds to q = 2)
I Energy of σ ∈ ΩN:
H(σ) =∑{i,j}⊂BN
i∼j
1{σi 6=σj}
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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Potts model
I Con�gurations: Let B(N) = {−N, . . . ,N}d.
ΩN ={σ = (σi)i∈B(N) : σi ∈ {1, . . . , q}
}(The Ising model corresponds to q = 2)
I Energy of σ ∈ ΩN:
H(σ) =∑{i,j}⊂BN
i∼j
1{σi 6=σj}
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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Potts model
I Con�gurations: Let B(N) = {−N, . . . ,N}d.
ΩN ={σ = (σi)i∈B(N) : σi ∈ {1, . . . , q}
}(The Ising model corresponds to q = 2)
I Energy of σ ∈ ΩN:
H(σ) =∑{i,j}⊂BN
i∼j
1{σi 6=σj}
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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Potts models
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) =1q
∀β > βc There is long-range order:
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:
ξβ(u) = − limn→∞
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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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:
ξβ(u) = − limn→∞
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.
20/35
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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:
ξβ(u) = − limn→∞
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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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:
ξβ(u) = − limn→∞
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.20/35
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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
I Energy of σ ∈ ΩN:
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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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.
23/35
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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.
23/35
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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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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
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
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
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 — 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)
(actually corresponds to the exact result by McCoy and Perk for the Ising model on Z2)
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Re�ned properties
?
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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Re�ned properties
Theorem (Ott, V. – arXiv:1706.09130)
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
Theorem (Ott, V. – arXiv:1706.09130)
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
Theorem (Ott, V. – arXiv:1706.09130)
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
Theorem (Ott, V. – arXiv:1706.09130)
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:
ξβ(Jc)− ξβ(J) ∼
(J− Jc)2 (d = 4)(J− Jc)/| log(J− Jc)| (d = 5)J− Jc (d ≥ 6)
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
Theorem (Ott, V. – arXiv:1706.09130)
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
Theorem (Ott, V. – arXiv:1706.09130)
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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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.
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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.
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Interface localization
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
n
All coupling constants are equal to 1 except those drawn in purple, whose the value isJ ≥ 0.
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Interface localization
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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Interface localization
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
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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
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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.
J. Bricmont and J. Fröhlich, Statistical mechanical methods in particle structureanalysis of lattice �eld theories. II. Scalar and surface models, Comm. Math. Phys.98 (1985), no. 4, 553–578.
W. J. Camp and M. E. Fisher, Behavior of two-point correlation functions at hightemperatures, Phys. Rev. Lett. 26 (1971), 73–77.
M. Campanino, D. Io�e, and Y. Velenik, Ornstein-Zernike theory for �nite rangeIsing models above Tc, Probab. Theory Related Fields 125 (2003), no. 3, 305–349.
, Random path representation and sharp correlations asymptotics athigh-temperatures, Adv. Stud. Pure Math. 39 (2004), 29–52.
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References ii
, Fluctuation theory of connectivities for subcritical random clustermodels, Ann. Probab. 36 (2008), no. 4, 1287–1321.
H. Duminil-Copin, A. Raou�, and V. Tassion, Sharp phase transition for therandom-cluster and Potts models via decision trees, preprint, arXiv:1705.03104.
R. A. Minlos and E. A. Zhizhina, Asymptotics of decay of correlations for latticespin �elds at high temperatures. I. The Ising model, J. Statist. Phys. 84 (1996),no. 1-2, 85–118.
S. Ott and Y. Velenik, Asymptotics of even-even correlations in the Ising model,preprint, arXiv:1804.08323.
, Potts models with a defect line, Commun. Math. Phys., to appear;arXiv:1706.09130.L. S. Ornstein and F. Zernike, Accidental deviations of density and opalescence atthe critical point of a single substance, Proc. Akad. Sci. 17 (1914), 793–806.
P. J. Paes-Leme, Ornstein-Zernike and analyticity properties for classical latticespin systems, Ann. Physics 115 (1978), no. 2, 367–387.
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References iii
A. M. Polyakov, Microscopic description of critical phenomena, J. Exp. Theor. Phys.28 (1969), no. 3, 533–539.
F. Zernike, The clustering-tendency of the molecules in the critical state and theextinction of light caused thereby, Koninklijke Nederlandse Akademie vanWetenschappen Proceedings Series B Physical Sciences 18 (1916), 1520–1527.
E. A. Zhizhina and R. A. Minlos, Asymptotics of the decay of correlations for Gibbsspin �elds, Teoret. Mat. Fiz. 77 (1988), no. 1, 3–12.
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