Height representation of XOR-Ising loops via bipartite dimers C - PowerPoint PPT Presentation

Height representation of XOR-Ising loops via bipartite dimers C´ edric Boutillier (UPMC) B´ eatrice de Tili` ere (UPMC) LAGA Universit´ e Paris Nord – March 11, 2015

The Ising model and the XOR-Ising model

The Ising model ◮ Let G = ( V , E ) be a finite graph embedded in the plane ◮ spin configuration σ : V − → {− 1 , +1 } ◮ σ assigns to every vertex x a spin σ x ∈ {− , + } +1 / − 1 are represented by green/blue dots.

The Ising model ◮ Edges of G are assigned positive coupling constants : J = ( J e ) e ∈ E . ◮ Ising Boltzmann measure :   1  � ∀ σ ∈ {− 1 , 1 } V ,  , P Ising ( σ ) = Z Ising ( G , J ) exp J xy σ x σ y e = xy ∈ E   �  �  is the where Z Ising ( G , J ) = exp J xy σ x σ y σ ∈{− 1 , 1 } V e = xy ∈ E Ising partition function .

The XOR-Ising model Ising model on G , J = ( Je ) e ∈ E Ising model on G , J = ( Je ) e ∈ E σ ′ σ × = × = × = ξ = σσ ′ × = XOR-Ising model on G , J = ( Je ) e ∈ E

The XOR-Ising model Ising model on G , J = ( Je ) e ∈ E Ising model on G , J = ( Je ) e ∈ E σ ′ σ × = × = × = ξ = σσ ′ × = XOR-Ising model on G , J = ( Je ) e ∈ E

Conjecture for the XOR-Ising model Conjecture (Wilson (11), Ikhlef–Picco–Santachiara) The scaling limit of polygon configurations separating ± 1 clusters of the critical XOR-Ising model are contour lines of the Gaussian √ free field, with the heights of the contours spaced 2 times as far apart as they are for [...] the double dimer model on the square lattice.

Result Theorem (B–dT) ◮ Polygon configurations of the XOR-Ising model have the same law as a family of contours in a bipartite dimer model. ◮ This family of contours are the level lines of a restriction of the height function of this bipartite dimer model. Remark Proved when the graph G is embedded in a surface of genus g , or when G is planar, infinite. ◮ When the XOR-Ising model is critical , so is the bipartite dimer model. ◮ Using results of [dT] on the convergence of the height function, this gives partial proof of Wilson’s conjecture.

Contour expansion of the Ising partition function [Kramers & Wannier]

Low temperature expansion ◮ Polygon configuration : subset of edges s.t. each vertex is incident to an even number of edges. e J e σ x σ y = e J e ( δ { σ x = σ y } + e − 2 J e δ { σ x � = σ y } ) . ◮ Write The partition function is then equal to (LTE) : e J e σ x σ y = C � � � � e − 2 J e . Z Ising ( G , J ) = P ∗ ∈ P ( G ∗ ) e ∗ ∈ P ∗ σ ∈{− 1 , 1 } V e = xy ∈ E ◮ Geometric interp: polygon config. separate clusters of ± 1 spins.

High temperature expansion e J e σ x σ y = cosh( J e )(1 + σ x σ y tanh( J e )) . ◮ Write, The partition function is then equal to (HTE) : e J e σ x σ y = C ′ � � � � Z Ising ( G , J ) = tanh( J e ) . σ ∈{− 1 , 1 } V e = xy ∈ E P ∈ P ( G ) e ∈ P ◮ No geometric interpretation using spin configurations.

Mixed contour expansion for the double Ising model

The double Ising model ◮ Take 2 independent copies (red/blue) of an Ising model on G, with coupling constants J . ◮ Using the LTE, consider the probability measure P 2 - Ising : if P ∗ , P ∗ are two polygon configurations. C 2 � � e ∗ ∈ P ∗ e − 2 J e �� � e ∗ ∈ P ∗ e − 2 J e � P 2 - Ising ( P ∗ , P ∗ ) = , Z 2 - Ising ( G , J ) where Z 2 - Ising ( G , J ) = Z Ising ( G , J ) 2 .

The double Ising model ◮ Let P ∗ , P ∗ be two polygon configurations. ◮ Consider the superimposition P ∗ ∪ P ∗ . ◮ Define two new edge configurations: ◮ Mono( P ∗ , P ∗ ) : monochromatic edges. ◮ Bi( P ∗ , P ∗ ) : bichromatic edges.

Monochromatic edges Monochromatic edge configuration of P ∗ ∪ P ∗ Lemma Mono( P ∗ , P ∗ ) is the polygon configuration separating ± 1 clusters of the corresponding XOR-Ising spin configuration. Goal: understand the law of monochromatic edge configurations.

Bichromatic edge configurations ◮ Let ( P ∗ , P ∗ ) be two polygon configurations. ◮ Mono( P ∗ , P ∗ ) splits the surface into connected comp. (Σ i ) i . Σ 9 Σ 8 Σ 1 Σ 2 Σ 3 Σ 7 Σ 6 Σ 4 Σ 5 Lemma For every i , the restriction of Bi( P ∗ , P ∗ ) to Σ i is the LTE of an Ising configuration on G Σ i , with coupling constants (2 J e ) .

Probability of monochromatic configurations Lemma Let P ∗ be a polygon configuration, separating the surface into n connected components. For every i , let P ∗ i be a polygon configuration of G ∗ Σ i . Then, there are 2 n pairs of polygon configurations ( P ∗ , P ∗ ) having P ∗ as monochromatic edges, and P ∗ 1 , · · · , P ∗ n as bichromatic edges. Denote by W ( P ∗ ) the contribution to Z 2 - Ising ( G , J ) of the pairs of polygon configurations ( P ∗ , P ∗ ) such that Mono( P ∗ , P ∗ ) = P ∗ . Corollary � � e ∗ ∈ P ∗ e − 2 J e � � n ◮ W ( P ∗ ) = C �� 2 Z LT ( G ∗ Σ i , 2 J ) i =1 ◮ Z 2 - Ising ( G , J ) = � P ∗ ∈ P ( G ∗ ) W ( P ∗ ) W ( P ∗ ) P 2 - Ising (Mono = P ∗ ) = Z 2 - Ising ( G ,J ) .

Mixed contour expansion � � W ( P ∗ ) = C e ∗ ∈ P ∗ e − 2 J e � � n 2 Z LT ( G ∗ �� Σ i , 2 J ) . i =1 Idea [Nienhuis]: high temperature expansion in each connected component Σ i . Z LT ( G ∗ Σ i , 2 J ) = C (Σ i ) Z HT ( G Σ i , 2 J ) . Low temp. expansion on G ∗ High temp. expansion on G Σ i . Σ i

Mixed contour expansion Proposition For every polygon configuration P ∗ , � 2 e − 2 J e � 1 − e − 4 J e � � � � � W ( P ∗ ) = C 1 + e − 4 J e 1 + e − 4 J e e ∗ ∈ P ∗ { P ∈ P ( G ): P ∗ ∩ P = ∅} e ∈ P 2 e − 2 Je 1 − e − 4 Je � � � � � � � 1+ e − 4 Je 1+ e − 4 Je e ∗∈ P ∗ { P ∈ P ( G ): P ∗∩ P = ∅} e ∈ P P 2 - Ising (Mono = P ∗ ) = � ··· P ∗∈ P ( G ∗ )

Higher genus If the graph is embedded in a surface Σ of genus g ≥ 0 . ◮ Consider H 1 (Σ , Z / 2 Z ) ≃ { 0 , 1 } 2 g . ◮ Family of Ising models, indexed by ε ∈ { 0 , 1 } 2 g . ◮ The double Ising model partition function is defined as: � Ising ( G , J ) 2 . Z ε Z 2 - Ising ( G , J ) = ε ∈{ 0 , 1 } 2 g

From mixed polygon configurations to dimers

The graph G Q = ( V Q , E Q )

The dimer model on G Q dimer configuration of G Q : a subset of edges M such that each vertex is incident to exactly on edge of M

The dimer model on G Q dimer configuration of G Q : a subset of edges M such that each vertex is incident to exactly on edge of M

The dimer model on G Q dimer configuration of G Q : a subset of edges M such that each vertex is incident to exactly on edge of M weight function ν on the edges Dimer Boltzmann measure : P dimer ( M ) ∝ � e ∈ E Q ν e

First step: from polygons to 6-vertex [Nienhuis] 1 2 3 4 5 6 Local mapping 1 2 3 4 5 6 1+ e − 4 Je , ω 34 = 1 − e − 4 Je 2 e − 2 Je Weights: ω 12 = 1+ e − 4 Je , ω 56 = 1 .

First step: from polygons to 6-vertex [Nienhuis] 1 2 3 4 5 6 Local mapping 1 2 3 4 5 6 1+ e − 4 Je , ω 34 = 1 − e − 4 Je 2 e − 2 Je Weights: ω 12 = 1+ e − 4 Je , ω 56 = 1 .

Second step: from 6V to dimers [Wu-Lin, Dub´ edat] 1 1 ω ω ω ω 12 12 34 34 1 ω 12 1 Local mapping ω 34 ω 1 2 3 4 5 6 34 ω 1 1 12 ω 12 ω 12 ω 34 ω 34 1 2 2 ω 12 + ω = 1 34

Conclusion ◮ To every dimer configuration M of G Q , assign Poly( M ) = (Poly 1 ( M ) , Poly 2 ( M )) , the pair of polygon configurations given by the mappings. Theorem For every polygon configuration P ∗ of G ∗ , P 2 - Ising (Mono = P ∗ ) = P dimer (Poly 1 = P ∗ )

Height function for bipartite dimers (Thurston)

Height function for bipartite dimers (Thurston) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Height function for bipartite dimers (Thurston) 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0