The double random current nesting field Marcin Lis University of - - PowerPoint PPT Presentation

The double random current nesting field

Marcin Lis

University of Cambridge

(joint work with Hugo Duminil-Copin) June 4, 2018

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Outline

◮ Three related (planar) models and their observables:

◮ the Ising model and the spontaneous magnetization ◮ the (double) random current model and the nesting field ◮ the dimer model and the height function

◮ I will then describe a measure preserving mapping between

double currents and dimers which maps the nesting field to the height function

◮ Application: I will use it and the results of Kenyon, Okounkov

and Sheffield to prove continuity of phase transition for Ising models on planar biperiodic graphs

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Ising model

Let Γ be a planar biperiodic graph, i.e., invariant under a ≃ Z2

• action. Let G = (V, E) be a finite connected subraph of Γ. Let U be

its set of faces and u∂ its unbounded face. Consider the space of spin configurations Σ =

• σ ∈ {−1, +1}U : σu∂ = +1
• .

The Ising model with “+” boundary conditions on the faces of G is a probability measure on Σ given by PG,β

Isg (σ) =

1 ZG,β

Isg

exp

• β
• u∼u′

J{u,u′}∗σuσu′

• ,

σ ∈ Σ, where {u, u′}∗ is the edge separating u and u∗, and the ferromagnetic coupling constants Je > 0 are biperiodic on the edges of Γ.

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Ising model - magnetization

We define the spontaneous magnetization at a face u σuΓ

β := lim GրΓ EG,β Isg [σu] 0.

The critical point is defined to be βc = sup{β > 0 : σuΓ

β = 0 for some u}.

How to find βc?

Cimasoni and Duminil-Copin (’13) computed βc as the only root of an explicit polynomial in xe = exp(−2βJe) depending on the local structure of Γ.

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Ising model - magnetization

What happens at βc? This is a question of continuity of phase

• transition. We will prove the following

Duminil-Copin & L. ’17

For any biperiodic graph Γ, σuΓ

βc = 0.

This is also a universality result. On the square lattice it was proved by Yang in 1952. On biperiodic isoradial graphs it follows from recent results of Chelkak, Hongler, Izyurov and Smirnov.

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Ising model - magnetization

Proof:

◮ It is enough to prove that σuΓ βc 0. ◮ Assume we can show that

PΓ,βc

Isg (u +1

← → ∞) = 0.

◮ Take a large box in Γ∗ containing u and condition on the

• utermost circuit of −1 faces. By the previous point, we

condition on an event of probability approaching 1.

◮ The conditional distribution inside the circuit is that of an Ising

model with −1 boundary conditions and hence contributes a nonpositive number to the magnetization.

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Ising model - contour representation

The space of even subgraphs is E = {ω ⊂ E : degv(ω) is even for all v ∈ V}. The Ising model is equivalent to a measure on E given by PG,β

Isg (ω) ∝

• e∈ω

xe, ω ∈ E, where xe = exp(−2βJe). A connected component of an even subgraph is called a contour.

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Random currents

A current is a function ω : E → {0, 1, 2} such that for every vertex v,

• e∋v ω(e) is even.

We write ωi = ω−1({i}) ⊆ E and we call ω1 odd and ω2 even edges respectively. Note that a function ω : E → {0, 1, 2} is a current iff ω1 ∈ E (!). Let Ω be the set of all currents, and let pe = 1 −

• 1 − x2
• e. The

random current probability measure is PG,β

curr(ω) =

1 ZG,β

curr

• e∈ω1

xe

• e∈ω2

pe

• e∈ω0

(1 − pe), ω ∈ Ω. We often identify a current ω with the set of edges with nonzero value

• f ω. A connected component of ω will be called a cluster.

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Random currents

Let xe = exp(−2βJe).

planar random current = Ising contours + percolation

Let ω1 be the Ising contour configuration, and let ω′

2 be Bernoulli

bond percolation with success probabilities pe. Then, ω1 and ω2 := ω′

2 \ ω1 define a random current configuration with

parameters xe. Our definition is derived directly from the original one of Griffiths, Hurst and Sherman (’70).

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Double random currents

The double random current probability measure is the measure of the sum of two i.i.d. random currents (which is again a current). PG,β

d-curr(ω) = PG,β curr ⊗ PG,β curr({(ω′, ω′′) ∈ Ω × Ω | ω′ + ω′′ = ω}),

ω ∈ Ω, where the sum ω′ + ω′′ is defined by the following table ω′\ω′′ 1 2 1 2 1 1 2 1 2 2 1 2 Note that ω1 is the contour configuration of the XOR Ising model corresponding to contours ω1

• dd and ω2
• dd.

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Double random currents

Proposition (L. ’16)

PG,β

d-curr(ω) =

1 ZG,β

d-curr

2|ω|+k(ω)

e∈ω1

xe

• e∈ω2

x2

e

• e∈ω0

(1 − x2

e)

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Double random currents

Double random currents posses a special combinatorial structure which is expressed in the celebrated switching lemma. It allows to study Ising correlation functions through percolation properties of random currents. They were used

◮ by Aizenman (’82) to prove triviality of the Ising field in

dimension d > 4.

◮ by Aizenman, Barsky and Fernandez (’87) to obtain sharpness of

phase transition for a general family of translation invariant spins systems,

◮ by Aizenman, Duminil-Copin and Sidoravicius (’14) to prove

continuity of phase transition for Ising models on a large family

• f lattices including Z3,

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Double random currents - nesting field

For each cluster C of a current ω, we toss an independent ±1 symmetric coin ξC . A cluster C is called odd around a face u if ω1 ∩ C interpreted as contours assigns spin −1 to u under +1 boundary conditions. The nesting field of ω at u is defined to be Su =

• C odd around u

ξC . The double random current nesting field is the law of S when ω is drawn according to PG,β

d-curr.

It can be thought of as a model of a random surface.

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Dimers

A perfect matching (or a dimer cover) M of a graph GD is a set of edges such that each vertex is incident on exactly one edge in M. Let M be the space of all perfect matchings. Let ze > 0, e ∈ E. The dimer model is a probability measure on M given by PGD,z

dim (M) =

1 ZGD,z

dim

• e∈M

ze, M ∈ M. The dimer model on bipartite graphs is a model of a random surface.

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Dimers - height function

Let GD be bipartite and color its vertices black and white. A flow is an antisymmetric function on the directed edges of GD. A perfect matching M can be identified with a flow with value +1 on every edge in M oriented from white to black, and value 0 on every edge not in M. For a reference matching M0, the flow M − M0 is a divergence free flow. The height function h is a function on the faces of GD defined in the following way:

◮ h(u∂) = 0, ◮ for other u, h(u) is the total flux of M − M0 across any path in

the dual of GD connecting u and u∂. Note: this is well defined.

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A measure preserving mapping

If the random current weights on G are xe, then the dimer weights on GD are zem =

2xe 1−x2

e for the middle edge and zes1 = zes2 = xe for the

side edges. Short edges get weight 1.

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A measure preserving mapping - edge factors

PG,β

d-curr(ω) ∼ 2|ω|+k(ω) e∈ω1

xe

• e∈ω2

x2

e

• e∈ω0

(1 − x2

e)

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A measure preserving mapping - edge factors

PG,β

d-curr(ω) ∼ 2|ω|+k(ω) e∈ω1

xe

• e∈ω2

x2

e

• e∈ω0

(1 − x2

e)

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A measure preserving mapping - edge factors

PG,β

d-curr(ω) ∼ 2|ω|+k(ω) e∈ω1

xe

• e∈ω2

x2

e

• e∈ω0

(1 − x2

e)

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A measure preserving mapping - vertex factors

PG,β

d-curr(ω) ∼ 2|ω|+k(ω) e∈ω1

xe

• e∈ω2

x2

e

• e∈ω0

(1 − x2

e)

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A measure preserving mapping - vertex factors

PG,β

d-curr(ω) ∼ 2|ω|+k(ω) e∈ω1

xe

• e∈ω2

x2

e

• e∈ω0

(1 − x2

e)

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A measure preserving mapping - cluster factors

PG,β

d-curr(ω) ∼ 2|ω|+k(ω) e∈ω1

xe

• e∈ω2

x2

e

• e∈ω0

(1 − x2

e)

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A measure preserving mapping - cluster factors

PG,β

d-curr(ω) ∼ 2|ω|+k(ω) e∈ω1

xe

• e∈ω2

x2

e

• e∈ω0

(1 − x2

e)

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A measure preserving mapping

Note that the set of faces of G embeds naturally in the set of faces of GD.

Theorem (Duminil-Copin & L., 2017)

Under the mapping the height function on GD restricted to the faces

• f G becomes the double random current nesting field S.

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Other maps between double Ising and dimers

Dub´ edat (2011) provided a mapping between the double Ising model

• n a graph G and dimers on a related graph CG.

Boutillier and de Tili` ere (2012) provided a different proof of the same mapping and showed convergence to full plane GFF.

However,

this map does not carry the structure of a nesting field, and it is more difficult to use information about dimers to study properties of Ising spins.

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Properties of the nesting field

For a face u, let Nu = Nu(ω) be the number of clusters of ω odd around u (nesting depth), and let Cu = Cu(ω) be the number of contours (i.e. connected components of ω1) in ω surrounding u. The crucial properties of N are: (1) EG,β

d-curr[Nu] = VarG,β d-curr[Su],

(2) Nu(ω1 + ω2) Nu(ω1) + Nu(ω2), and hence EG,β

Isg [Cu] EG,β curr[Nu] 1

2EG,β

d-curr[Nu] = 1

2VarG,β

d-curr[Su].

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Asymptotics of the dimer height function

Theorem (Kenyon–Okounkov–Sheffield, ’03)

If the dimer model is in the liquid phase (local condition in the weights z = zc), then VarΓD,zc

dim [h(u) − h(u′)] = 1

π log |u − u′|, as |u − u′| → ∞. Remark: It would be enough for us to know that the variance diverges with no exact asymptotics.

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Proof of vanishing magnetization

(1) by Cimasoni–Duminil-Copin (’13), criticality of Ising on Γ is equivalent to liquid phase of dimers on ΓD, (2) by Kenyon–Okounkov–Sheffield (’03), for liquid dimer models on bipartite biperiodic graphs and periodic boundary conditions, VarΓD,zc

dim [h(u) − h(u′)] → ∞

as |u − u′| → ∞, (3) by Raoufi (’17), the periodic boundary conditions state is a convex combination of pure states (4) hence, by our mapping and properties of nesting depth, EΓ,βc

Isg [Cu] 1

2EΓD,βc

d-curr[Nu] = 1

2 lim

|u′|→∞

VarΓ,zc

dim [h(u) − h(u′)] = ∞.

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Proof of vanishing magnetization - (5)

Let P = PΓ,β

Isg and E = EΓ,β Isg .

Lemma

If E[Cu] = ∞, then P(u

+1

← → ∞) = 0. Proof by contradiction:

◮ Assume that p = P(u +1

← → ∞) > 0.

◮ We wish to prove that for every k 0,

P[Cu(σ) k + 2 | Cu(σ) k] 1 − p. This immediately implies that E[Cu(σ)] < ∞.

◮ Note that it suffices to show that for every k 0,

P[Cu(σ) 2k + 1 | Cu(σ) 2k and σu = +] 1 − p, P[Cu(σ) 2k + 2 | Cu(σ) 2k + 1 and σu = −] 1 − p. .

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Proof of vanishing magnetization - (5)

◮ For k 1, let F be the set of faces v of Γ for which every path

from u to v contains at least 2k changes of signs.

◮ Fix a set of faces F. For

σ ∈ AF := {F = F} ∩ {Cu 2k} ∩ {σu = +1}, faces on ∂F have spin +1. Therefore, by spatial Markov property P[Cu 2k + 1|AF] P[∂F

+1

← → ∞|AF] = 1 − P[∂F

+1

← → ∞ | σv = +1 for all v ∈ ∂F].

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Proof of vanishing magnetization - (5)

◮ Note that {u +1

← → ∞} ⊆ {∂F

+1

← → ∞}. Hence, by FKG: P[∂F

+1

← →∞|σv = +1 for all v ∈ ∂F] P[∂F

+1

← → ∞] P[u

+1

← → ∞] p, so that P[Cu 2k + 1|AF] 1 − p. We finish the proof by noticing that AF partition {Cu(σ) 2k} ∩ {σu = +1}.

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Thank you for your attention!

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