The double random current nesting field Marcin Lis University of Cambridge (joint work with Hugo Duminil-Copin) June 4, 2018 1 / 32
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 2 / 32
Ising model Let Γ be a planar biperiodic graph, i.e., invariant under a ≃ Z 2 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 ∈ {− 1 , +1 } U : σ u ∂ = +1 � � Σ = . The Ising model with “+” boundary conditions on the faces of G is a probability measure on Σ given by 1 � � P G ,β � Isg ( σ ) = exp J { u , u ′ } ∗ σ u σ u ′ σ ∈ Σ , β , Z G ,β Isg u ∼ u ′ where { u , u ′ } ∗ is the edge separating u and u ∗ , and the ferromagnetic coupling constants J e > 0 are biperiodic on the edges of Γ. 3 / 32
Ising model - magnetization We define the spontaneous magnetization at a face u G ր Γ E G ,β � σ u � Γ β := lim Isg [ σ u ] � 0 . The critical point is defined to be β c = sup { β > 0 : � σ u � Γ β = 0 for some u } . How to find β c ? Cimasoni and Duminil-Copin (’13) computed β c as the only root of an explicit polynomial in x e = exp( − 2 β J e ) depending on the local structure of Γ. 4 / 32
Ising model - magnetization What happens at β c ? This is a question of continuity of phase transition. We will prove the following Duminil-Copin & L. ’17 For any biperiodic graph Γ, � σ u � Γ β c = 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. 5 / 32
Ising model - magnetization Proof: ◮ It is enough to prove that � σ u � Γ β c � 0. ◮ Assume we can show that +1 P Γ ,β c Isg (u ← → ∞ ) = 0 . ◮ Take a large box in Γ ∗ containing u and condition on the outermost 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. � 6 / 32
Ising model - contour representation The space of even subgraphs is E = { ω ⊂ E : deg v ( ω ) is even for all v ∈ V } . The Ising model is equivalent to a measure on E given by P G ,β � Isg ( ω ) ∝ x e , ω ∈ E , e ∈ ω where x e = exp( − 2 β J e ). A connected component of an even subgraph is called a contour. 7 / 32
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 ω i = ω − 1 ( { i } ) ⊆ E and we call ω 1 odd and ω 2 even edges respectively. Note that a function ω : E → { 0 , 1 , 2 } is a current iff ω 1 ∈ E (!). � 1 − x 2 Let Ω be the set of all currents, and let p e = 1 − e . The random current probability measure is 1 � � � P G ,β curr ( ω ) = x e p e (1 − p e ) , ω ∈ Ω . Z G ,β curr e ∈ ω 1 e ∈ ω 2 e ∈ ω 0 We often identify a current ω with the set of edges with nonzero value of ω . A connected component of ω will be called a cluster. 8 / 32
Random currents Let x e = exp( − 2 β J e ). planar random current = Ising contours + percolation Let ω 1 be the Ising contour configuration, and let ω ′ 2 be Bernoulli bond percolation with success probabilities p e . Then, ω 1 and ω 2 := ω ′ 2 \ ω 1 define a random current configuration with parameters x e . Our definition is derived directly from the original one of Griffiths, Hurst and Sherman (’70). 9 / 32
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). curr ( { ( ω ′ , ω ′′ ) ∈ Ω × Ω | ω ′ + ω ′′ = ω } ) , P G ,β d-curr ( ω ) = P G ,β curr ⊗ P G ,β ω ∈ Ω , where the sum ω ′ + ω ′′ is defined by the following table ω ′ \ ω ′′ 0 1 2 0 0 1 2 1 1 2 1 2 2 1 2 Note that ω 1 is the contour configuration of the XOR Ising model corresponding to contours ω 1 odd and ω 2 odd . 10 / 32
Double random currents Proposition (L. ’16) 1 P G ,β 2 | ω | +k( ω ) � � � x 2 (1 − x 2 d-curr ( ω ) = x e e ) e Z G ,β d-curr e ∈ ω 1 e ∈ ω 2 e ∈ ω 0 11 / 32
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 of lattices including Z 3 , 12 / 32
Double random currents - nesting field For each cluster C of a current ω , we toss an independent ± 1 symmetric coin ξ C . A cluster C is called odd around a face u if ω 1 ∩ C interpreted as contours assigns spin − 1 to u under +1 boundary conditions. The nesting field of ω at u is defined to be � S u = ξ C . C odd around u The double random current nesting field is the law of S when ω is drawn according to P G ,β d-curr . It can be thought of as a model of a random surface. 13 / 32
Dimers A perfect matching (or a dimer cover) M of a graph G D 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 z e > 0, e ∈ E. The dimer model is a probability measure on M given by 1 P G D , z � dim (M) = z e , M ∈ M . Z G D , z dim e ∈ M The dimer model on bipartite graphs is a model of a random surface. 14 / 32
Dimers - height function Let G D be bipartite and color its vertices black and white. A flow is an antisymmetric function on the directed edges of G D . 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 M 0 , the flow M − M 0 is a divergence free flow. The height function h is a function on the faces of G D defined in the following way: ◮ h(u ∂ ) = 0, ◮ for other u, h(u) is the total flux of M − M 0 across any path in the dual of G D connecting u and u ∂ . Note: this is well defined. 15 / 32
A measure preserving mapping If the random current weights on G are x e , then the dimer weights on 2x e G D are z e m = e for the middle edge and z e s1 = z e s2 = x e for the 1 − x 2 side edges. Short edges get weight 1. 16 / 32
A measure preserving mapping - edge factors P G ,β d-curr ( ω ) ∼ 2 | ω | +k( ω ) � � � x 2 (1 − x 2 x e e ) e e ∈ ω 1 e ∈ ω 2 e ∈ ω 0 17 / 32
A measure preserving mapping - edge factors P G ,β d-curr ( ω ) ∼ 2 | ω | +k( ω ) � � � x 2 (1 − x 2 x e e ) e e ∈ ω 1 e ∈ ω 2 e ∈ ω 0 18 / 32
A measure preserving mapping - edge factors P G ,β d-curr ( ω ) ∼ 2 | ω | +k( ω ) � � � x 2 (1 − x 2 x e e ) e e ∈ ω 1 e ∈ ω 2 e ∈ ω 0 19 / 32
A measure preserving mapping - vertex factors P G ,β d-curr ( ω ) ∼ 2 | ω | +k( ω ) � � � x 2 (1 − x 2 x e e ) e e ∈ ω 1 e ∈ ω 2 e ∈ ω 0 20 / 32
A measure preserving mapping - vertex factors P G ,β d-curr ( ω ) ∼ 2 | ω | +k( ω ) � � � x 2 (1 − x 2 x e e ) e e ∈ ω 1 e ∈ ω 2 e ∈ ω 0 21 / 32
A measure preserving mapping - cluster factors P G ,β d-curr ( ω ) ∼ 2 | ω | +k( ω ) � � � x 2 (1 − x 2 x e e ) e e ∈ ω 1 e ∈ ω 2 e ∈ ω 0 22 / 32
A measure preserving mapping - cluster factors P G ,β d-curr ( ω ) ∼ 2 | ω | +k( ω ) � � � x 2 (1 − x 2 x e e ) e e ∈ ω 1 e ∈ ω 2 e ∈ ω 0 23 / 32
A measure preserving mapping Note that the set of faces of G embeds naturally in the set of faces of G D . Theorem (Duminil-Copin & L., 2017) Under the mapping the height function on G D restricted to the faces of G becomes the double random current nesting field S . 24 / 32
Other maps between double Ising and dimers Dub´ edat (2011) provided a mapping between the double Ising model on a graph G and dimers on a related graph C G . 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. 25 / 32
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