[PPT] - Order - disorder operators in planar and almost planar graphs (2) PowerPoint Presentation

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Order - disorder operators in planar and almost planar graphs (2)

Hugo Duminil-Copin, I.H.´ E.S.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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The main statement Consider the Ising’s Hamiltonian (with free boundary conditions) on a finite subgraph G of the square lattice Z2 with coupling constants Jxy ≥ 0, HG(σ)

def

= −

x,y∈G

Jxyσxσy and the associated measure at inverse-temperature β defined for any f , f G,β =

σ∈{±1}G

f (σ) exp[−βHG(σ)]

σ∈{±1}G

exp[−βHG(σ)] .

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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The main statement Consider the Ising’s Hamiltonian (with free boundary conditions) on a finite subgraph G of the square lattice Z2 with coupling constants Jxy ≥ 0, HG(σ)

def

= −

x,y∈G

Jxyσxσy and the associated measure at inverse-temperature β defined for any f , f G,β =

σ∈{±1}G

f (σ) exp[−βHG(σ)]

σ∈{±1}G

exp[−βHG(σ)] . Focus on symmetric finite range (i.e. Jxy = 0 for |x − y| ≥ R) Ising model at the critical inverse temperature βc on the upper half-plane H.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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The main statement Consider the Ising’s Hamiltonian (with free boundary conditions) on a finite subgraph G of the square lattice Z2 with coupling constants Jxy ≥ 0, HG(σ)

def

= −

x,y∈G

Jxyσxσy and the associated measure at inverse-temperature β defined for any f , f G,β =

σ∈{±1}G

f (σ) exp[−βHG(σ)]

σ∈{±1}G

exp[−βHG(σ)] . Focus on symmetric finite range (i.e. Jxy = 0 for |x − y| ≥ R) Ising model at the critical inverse temperature βc on the upper half-plane H. Theorem (Aizenman, Duminil-Copin, Tassion, Warzel (2016)) For x1, . . . , x2n found in this order on the boundary of H, σx1 · · · σx2nH,βc ∼ Pfaff

σxi σxj H,βc
1≤i<j≤2n
as min |xi − xj| tends to infinity.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Part I. Planar Case

Rewriting correlations functions in terms of random currents Rewrite the model in terms of integer-valued functions (called currents) m = (mxy : x, y ∈ G).

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Part I. Planar Case

Rewriting correlations functions in terms of random currents Rewrite the model in terms of integer-valued functions (called currents) m = (mxy : x, y ∈ G). As observed by Griffiths, Hurst and Sherman (1970), the identity exp[βJxyσxσy] =

∞

mxy =0

(βJxyσxσy)mxy mxy! allows us to write for σA =

x∈A σx,

σ∈{±1}G

σA exp[−βHG(σ)]

def

=

σ∈{±1}G

σA

x,y∈G

exp[βJxyσxσy]

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 7

Part I. Planar Case

Rewriting correlations functions in terms of random currents Rewrite the model in terms of integer-valued functions (called currents) m = (mxy : x, y ∈ G). As observed by Griffiths, Hurst and Sherman (1970), the identity exp[βJxyσxσy] =

∞

mxy =0

(βJxyσxσy)mxy mxy! allows us to write for σA =

x∈A σx,

σ∈{±1}G

σA exp[−βHG(σ)]

switch sums

=

m

w(m)

σ∈{±1}G
x∈G

σ I[x∈A]+∆(m,x)

x

where w(m)

def

=

x∼y

βmxy

mxy ! and ∆(m, x) def

=

y mxy.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 8

Part I. Planar Case

Rewriting correlations functions in terms of random currents Rewrite the model in terms of integer-valued functions (called currents) m = (mxy : x, y ∈ G). As observed by Griffiths, Hurst and Sherman (1970), the identity exp[βJxyσxσy] =

∞

mxy =0

(βJxyσxσy)mxy mxy! allows us to write for σA =

x∈A σx,

σ∈{±1}G

σA exp[−βHG(σ)] = 2|G|

∂m=A

w(m), where w(m)

def

=

x∼y

βmxy

mxy ! and sources ∂m def

= {x ∈ G, ∆(m, x) odd}. based on the fact that for each fixed x ∈ G, the map flipping σx is an involution on spin configurations.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

1 2 3 1 2 2 1 5 1 2

n a b a M b

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 13

An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 14

An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 18

An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 20

An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 21

An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 23

An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 24

An interpretation of currents in terms of loops Identify m = (mxy : x, y ∈ G) with a (multi-)graph M with mxy edges between x and y.

a M b

A current m with sources ∂m = A can be seen as a collection of loops together with paths pairing the vertices of A together.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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The switching principle and its applications Lemma (Switching lemma) For A ⊂ G and x, y ∈ G

∂m1=A

∂m2={x,y}

w(m1)w(m2) =

∂m1=A∆{x,y}

∂m2=∅

w(m1)w(m2)I[x

M1∪M2

← → y].

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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The switching principle and its applications Lemma (Switching lemma) For A ⊂ G and x, y ∈ G

∂m1=A

∂m2={x,y}

w(m1)w(m2) =

∂m1=A∆{x,y}

∂m2=∅

w(m1)w(m2)I[x

M1∪M2

← → y]. Let F be a function of two currents, then

∂m1=B

∂m2=A

F(m1, m2)w(m1)w(m2) =

∂m=A∆B

w(m)

∂n=B

n≤m

F(n, m − n)

m

n

.

Simply make the change of variables m = m1 + m2 and n = m1, and observe that w(m1)w(m2) = w(m) m

n

where

m

n

:=

x,y

mxy

nxy

Hugo Duminil-Copin, I.H.´

E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 27

The switching principle and its applications Lemma (Switching lemma) For A ⊂ G and x, y ∈ G

∂m1=A

∂m2={x,y}

w(m1)w(m2) =

∂m1=A∆{x,y}

∂m2=∅

w(m1)w(m2)I[x

M1∪M2

← → y]. Let F be a function of two currents, then

∂m1=B

∂m2=A

F(m1, m2)w(m1)w(m2) =

∂m=A∆B

w(m)

∂N =B

N ⊂M

F(N, M \ N). Simply make the change of variables m = m1 + m2 and n = m1, and observe that w(m1)w(m2) = w(m) m

n

where

m

n

:=

x,y

mxy

nxy

is the number of

subgraphs N of M with exactly nxy edges between each x and y.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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The switching principle and its applications Lemma (Switching lemma) For A ⊂ G and x, y ∈ G

∂m1=A

∂m2={x,y}

w(m1)w(m2) =

∂m1=A∆{x,y}

∂m2=∅

w(m1)w(m2)I[x

M1∪M2

← → y]. Let F be a function of two currents, then

∂m1=B

∂m2=A

F(m1, m2)w(m1)w(m2) =

∂m=A∆B

w(m)

∂N =B

N ⊂M

F(N, M \ N). Simply make the change of variables m = m1 + m2 and n = m1, and observe that w(m1)w(m2) = w(m) m

n

where

m

n

:=

x,y

mxy

nxy

is the number of

subgraphs N of M with exactly nxy edges between each x and y.

∂m=A∆{x,y}

w(m)

∂N ={x,y}

N ⊂M

1

?

=

∂m=A∆{x,y}

w(m)

∂N =∅

N ⊂M

I[x

M

← → y]

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 29

The switching principle and its applications Lemma (Switching lemma) For A ⊂ G and x, y ∈ G

∂m1=A

∂m2={x,y}

w(m1)w(m2) =

∂m1=A∆{x,y}

∂m2=∅

w(m1)w(m2)I[x

M1∪M2

← → y]. Let F be a function of two currents, then

∂m1=B

∂m2=A

F(m1, m2)w(m1)w(m2) =

∂m=A∆B

w(m)

∂N =B

N ⊂M

F(N, M \ N). Simply make the change of variables m = m1 + m2 and n = m1, and observe that w(m1)w(m2) = w(m) m

n

where

m

n

:=

x,y

mxy

nxy

is the number of

subgraphs N of M with exactly nxy edges between each x and y.

∂m=A∆{x,y}

w(m)

∂N ={x,y}

N ⊂M

1

?

=

∂m=A∆{x,y}

w(m)

∂N =∅

N ⊂M

I[x

M

← → y]

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 30

The switching principle and its applications Lemma (Switching lemma) For A ⊂ G and x, y ∈ G

∂m1=A

∂m2={x,y}

w(m1)w(m2) =

∂m1=A∆{x,y}

∂m2=∅

w(m1)w(m2)I[x

M1∪M2

← → y]. Let F be a function of two currents, then

∂m1=B

∂m2=A

F(m1, m2)w(m1)w(m2) =

∂m=A∆B

w(m)

∂N =B

N ⊂M

F(N, M \ N). Simply make the change of variables m = m1 + m2 and n = m1, and observe that w(m1)w(m2) = w(m) m

n

where

m

n

:=

x,y

mxy

nxy

is the number of

subgraphs N of M with exactly nxy edges between each x and y.

∂m=A∆{x,y}

w(m)

∂N ={x,y}

N ⊂M

1

?

=

∂m=A∆{x,y}

w(m)

∂N =∅

N ⊂M

I[x

M

← → y]

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Applications of the switching principle σxσy2 =

∂m1=∂m2={x,y} w(m1)w(m2)
∂m1=∂m2=∅ w(m1)w(m2)

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 32

Applications of the switching principle σxσy2 =

∂m1=∂m2={x,y} w(m1)w(m2)
∂m1=∂m2=∅ w(m1)w(m2)

switching

=

∂m1=∂m2=∅ w(m1)w(m2)I[x

M1∪M2

← → y]

∂m1=∂m2=∅ w(m1)w(m2)

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 33

Applications of the switching principle σxσy2 =

∂m1=∂m2={x,y} w(m1)w(m2)
∂m1=∂m2=∅ w(m1)w(m2)

switching

=

∂m1=∂m2=∅ w(m1)w(m2)I[x

M1∪M2

← → y]

∂m1=∂m2=∅ w(m1)w(m2)

= P[x ← → y]. The square of spin-spin correlations can be interpreted using connection probabilities in a (highly dependent) percolation model. This explains why many bounds obtained for Bernoulli percolation work also for the square of the spin correlations in Ising (e.g. m∗(β) ≥ c√β − βc).

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Applications of the switching principle σxσy2 =

∂m1=∂m2={x,y} w(m1)w(m2)
∂m1=∂m2=∅ w(m1)w(m2)

switching

=

∂m1=∂m2=∅ w(m1)w(m2)I[x

M1∪M2

← → y]

∂m1=∂m2=∅ w(m1)w(m2)

= P[x ← → y]. The square of spin-spin correlations can be interpreted using connection probabilities in a (highly dependent) percolation model. This explains why many bounds obtained for Bernoulli percolation work also for the square of the spin correlations in Ising (e.g. m∗(β) ≥ c√β − βc). σAσxσy

switching

=

∂m1=A∆{x,y},∂m2=∅ w(m1)w(m2)I[x

M1∪M2

← → y]

∂m1=∂m2=∅ w(m1)w(m2)

= σA∆{x,y}

∂m1=A∆{x,y},∂m2=∅ w(m1)w(m2)I[x

M1∪M2

← → y]

∂m1=A∆{x,y},∂m2=∅ w(m1)w(m2)

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Applications of the switching principle σxσy2 =

∂m1=∂m2={x,y} w(m1)w(m2)
∂m1=∂m2=∅ w(m1)w(m2)

switching

=

∂m1=∂m2=∅ w(m1)w(m2)I[x

M1∪M2

← → y]

∂m1=∂m2=∅ w(m1)w(m2)

= P[x ← → y]. The square of spin-spin correlations can be interpreted using connection probabilities in a (highly dependent) percolation model. This explains why many bounds obtained for Bernoulli percolation work also for the square of the spin correlations in Ising (e.g. m∗(β) ≥ c√β − βc). σAσxσy

switching

=

∂m1=A∆{x,y},∂m2=∅ w(m1)w(m2)I[x

M1∪M2

← → y]

∂m1=∂m2=∅ w(m1)w(m2)

= σA∆{x,y} P[x ← → y].

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 36

Proof of the Pfaffian formula for boundary spin correlations For n = 2, see blackboard.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Proof of the Pfaffian formula for boundary spin correlations For n = 2, see blackboard. We prove the result by induction. For n ≥ 3, Pfaffn(A) =

2n

ℓ=2

(−1)ℓA1,ℓ Pfaffn−1([A]1,ℓ), so that it is sufficient to prove that

2n

ℓ=2

(−1)ℓσx1σxℓ

1≤j≤2n

j / ∈{1,ℓ}

σxj ? = σx1 · · · σx2n.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 38

Proof of the Pfaffian formula for boundary spin correlations For n = 2, see blackboard. We prove the result by induction. For n ≥ 3, Pfaffn(A) =

2n

ℓ=2

(−1)ℓA1,ℓ Pfaffn−1([A]1,ℓ), so that it is sufficient to prove that

2n

ℓ=2

(−1)ℓσx1σxℓ

1≤j≤2n

j / ∈{1,ℓ}

σxj ? = σx1 · · · σx2n. Using random-currents, we obtain LHS = σx1 · · · σx2n E

2n
ℓ=2

(−1)ℓ I[x1 ← → xℓ]

.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 39

Proof of the Pfaffian formula for boundary spin correlations For n = 2, see blackboard. We prove the result by induction. For n ≥ 3, Pfaffn(A) =

2n

ℓ=2

(−1)ℓA1,ℓ Pfaffn−1([A]1,ℓ), so that it is sufficient to prove that

2n

ℓ=2

(−1)ℓσx1σxℓ

1≤j≤2n

j / ∈{1,ℓ}

σxj ? = σx1 · · · σx2n. Using random-currents, we obtain LHS = σx1 · · · σx2n E

2n
ℓ=2

(−1)ℓ I[x1 ← → xℓ]

.

For a fixed percolation configuration, the prescribed source constraints implies that the sites xℓ for which x1 ← → xℓ have labels of alternating parity due to the planarity of the graph. Thus

2n

ℓ=2

(−1)ℓ I[x1 ← → xℓ] = 1 .

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Part II. Finite-range interactions

Heuristic in the case of finite-range interactions Let us focus on the four-point function. The representation in random-current still works, so that it would be sufficient to study intersections.

x1 x2 x3 x4

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 41

Part II. Finite-range interactions

Heuristic in the case of finite-range interactions Let us focus on the four-point function. The representation in random-current still works, so that it would be sufficient to study intersections.

x1 x2 x3 x4

It is no longer true that paths necessarily intersect.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 42

Part II. Finite-range interactions

Heuristic in the case of finite-range interactions Let us focus on the four-point function. The representation in random-current still works, so that it would be sufficient to study intersections.

x1 x2 x3 x4

It is no longer true that paths necessarily intersect. We will use the first current (the one with sources) to locate an avoided

intersection. Then, we use the second current to connect the two clusters of

the first current.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 43

Part II. Finite-range interactions

Heuristic in the case of finite-range interactions Let us focus on the four-point function. The representation in random-current still works, so that it would be sufficient to study intersections.

x1 x2 x3 x4

It is no longer true that paths necessarily intersect. We will use the first current (the one with sources) to locate an avoided

intersection. Then, we use the second current to connect the two clusters of

the first current. Theorem (Aizenman, Duminil-Copin, Tassion, Warzel (2016)) At βc, the (infinite-volume sourceless) random current contains infinitely many circuits surrounding the origin almost surely.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Three models related to Ising on a finite graph G In this part, let us assume (to simplify), that Jxy = 1 if {x, y} is an edge of G, and 0 otherwise. RC percolation. Model of random subgraph of G obtained by taking the trace

m of a sourceless current m sampled with probability proportional to w(m).

Loop O(1) model. Model of random even subgraph η of G obtained from the high-temperature expansion of the model (the probability is proportional to tanh(β)|η|. For G planar, the loops correspond to interfaces of the Ising model on G ∗ by Kramers-Wannier duality. FK-Ising percolation. Model of random subgraph ω of G, where φ(ω) := 1 Z

p

1 − p #edges in ωq#connected components in ω with p = 1 − e−2β and q = 2.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Coupling between these models

independent sprinking

FK-Ising RC-perco Loop O(1)

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Coupling between these models

independent sprinking independent sprinking

FK-Ising RC-perco Loop O(1)

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Coupling between these models

independent sprinking independent sprinking uniform even subgraph

FK-Ising RC-perco Loop O(1)

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Synergy between models

1. RC percolation. Particularly useful when working with truncated correlation

functions, especially because of the switching lemma.

2. Loop O(1)-model. Rich combinatorial structure due to the constraints on

configurations being even subgraphs. Additional switching principles. Also, in the planar case interpretation in terms of interfaces.

3. FK-Ising. FKG measure: the model is positively associated. In particular,
ne can prove a bunch of general theorems on the critical phase.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Synergy between models

1. RC percolation. Particularly useful when working with truncated correlation

functions, especially because of the switching lemma.

2. Loop O(1)-model. Rich combinatorial structure due to the constraints on

configurations being even subgraphs. Additional switching principles. Also, in the planar case interpretation in terms of interfaces.

3. FK-Ising. FKG measure: the model is positively associated. In particular,
ne can prove a bunch of general theorems on the critical phase.

Pseudo-Theorem At criticality, there is no infinite connected component in ω almost surely.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Synergy between models

1. RC percolation. Particularly useful when working with truncated correlation

functions, especially because of the switching lemma.

2. Loop O(1)-model. Rich combinatorial structure due to the constraints on

configurations being even subgraphs. Additional switching principles. Also, in the planar case interpretation in terms of interfaces.

3. FK-Ising. FKG measure: the model is positively associated. In particular,
ne can prove a bunch of general theorems on the critical phase.

Pseudo-Theorem At criticality, there is no infinite connected component in ω almost surely. Pseudo-Theorem There are infinitely many circuits around the origin in ω almost surely.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Heuristic proof for finite-range interactions The two previous theorems on FK-Ising show that there are infinitely many distinct connected components surrounding the origin almost surely. We now wish to prove that in a uniformly chosen even subgraph, there is infinitely many circuits surrounding the origin.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Heuristic proof for finite-range interactions The two previous theorems on FK-Ising show that there are infinitely many distinct connected components surrounding the origin almost surely. We now wish to prove that in a uniformly chosen even subgraph, there is infinitely many circuits surrounding the origin.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Heuristic proof for finite-range interactions The two previous theorems on FK-Ising show that there are infinitely many distinct connected components surrounding the origin almost surely. We now wish to prove that in a uniformly chosen even subgraph, there is infinitely many circuits surrounding the origin.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Heuristic proof for finite-range interactions The two previous theorems on FK-Ising show that there are infinitely many distinct connected components surrounding the origin almost surely. We now wish to prove that in a uniformly chosen even subgraph, there is infinitely many circuits surrounding the origin.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Conclusion One step towards universality: the behavior at criticality is expected to be independent of the local definition. It should depend on The ± symmetry of the spins, The global geometry of the lattice (planarity, dimension, growth, etc).

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Conclusion One step towards universality: the behavior at criticality is expected to be independent of the local definition. It should depend on The ± symmetry of the spins, The global geometry of the lattice (planarity, dimension, growth, etc). The proof is precisely based on these two ingredients: Random-Current representation relies on the ± symmetry of the spins,

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Conclusion One step towards universality: the behavior at criticality is expected to be independent of the local definition. It should depend on The ± symmetry of the spins, The global geometry of the lattice (planarity, dimension, growth, etc). The proof is precisely based on these two ingredients: Random-Current representation relies on the ± symmetry of the spins, The study of crossings on the planarity.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

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Conclusion One step towards universality: the behavior at criticality is expected to be independent of the local definition. It should depend on The ± symmetry of the spins, The global geometry of the lattice (planarity, dimension, growth, etc). The proof is precisely based on these two ingredients: Random-Current representation relies on the ± symmetry of the spins, The study of crossings on the planarity. Further applications to Ising are expected, e.g. to Order/Disorder and Energy correlations, or to exponents.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)

SLIDE 62

Conclusion One step towards universality: the behavior at criticality is expected to be independent of the local definition. It should depend on The ± symmetry of the spins, The global geometry of the lattice (planarity, dimension, growth, etc). The proof is precisely based on these two ingredients: Random-Current representation relies on the ± symmetry of the spins, The study of crossings on the planarity. Further applications to Ising are expected, e.g. to Order/Disorder and Energy correlations, or to exponents. The coupling generalizes to Ashkin-Teller models and has new applications there.

Hugo Duminil-Copin, I.H.´ E.S. Order - disorder operators in planar and almost planar graphs (2)