The critical Z -invariant Ising model via dimers B eatrice de - PowerPoint PPT Presentation

The critical Z -invariant Ising model via dimers B´ eatrice de Tili` ere University of Neuchˆ atel joint work with C´ edric Boutillier, University Paris 6 Ascona, 27 May 2010

The 2 -dimensional Ising model • Planar graph, G = ( V ( G ) , E ( G )). • Spin Configurations, σ : V ( G ) → {− 1 , 1 } . • Every edge e ∈ E ( G ) has a coupling constant, J e > 0. • Ising Boltzmann measure (finite graph):   1 �  , P Ising ( σ ) = exp J e σ u σ v  Z Ising e = uv ∈ E ( G ) where Z Ising is the partition function.

Isoradial graphs • Graph G isoradial : can be embedded in the plane so that every face is inscribable in a circle of radius 1. • G isoradial ⇒ G ∗ isoradial : vertices of G ∗ = circumcenters.

Isoradial graphs � vertices : V ( G ) ∪ V ( G ∗ ) • G ⋄ : corresponding diamond graph edges : radii of circles � rhombus • Edge e → e angle θ e θ e ⇒ Coupling constants: J e ( θ e ).

The Z -invariant Ising model [Baxter] • Star-triangle transformation on G : preserves isoradiality. • Z -invariant Ising model : satisfies ∆ − Y � 2 K ( k ) � sinh(2 J e ( θ e )) = sn θ e | k k 2 ∈ R . π ⇒ � , � 2 K ( k ) cn θ e | k π sn, cn : Jacobi elliptic trigonometric functions. K : complete elliptic integral of the first kind.

The critical Z -invariant Ising model [Baxter] • High and low temperature expansion of the partition function ⇒ measure on contour configurations of G and G ∗ . • Generalized form of self-duality ⇒ k = 0, and J e ( θ e ) = 1 � 1 + sin θ e � 2 log . (1) cos θ e • Examples. √ ◦ G = Z 2 : θ e = π/ 4, J e ( θ e ) = log � 1 + 2. 1 4 ). ◦ G triangular lattice : θ e = π/ 6, J e ( θ e ) = log(3 √ � ◦ G hexagonal lattice : θ e = π/ 3, J e ( θ e ) = log 2 + 3. Critical temperatures (Kramers, Kramers-Wannier) (1) are called critical coupling constants.

Statistical mechanics on isoradial graphs • Critical Ising model: [Baxter, Costa-Santos, Mercat, Smirnov, Chelkak & Smirnov]. • Explicit expressions for: ◦ the Green’s function with weights “tan( θ e )” [Kenyon], ◦ the inverse of the Dirac operator ¯ ∂ for bipartite graphs, with weights “2 sin θ e ” [Kenyon], which only depend on the local geometry of the graph. • What is special about this setting ? ◦ Z -invariance (integrability). ◦ Natural setting for discrete complex analysis [Duffin, Mercat, Chelkak & Smirnov].

The dimer model • Graph G = ( V ( G ) , E ( G )) (planar). • Dimer configuration: perfect matching M . • Weight function, ν : E ( G ) → R + • Dimer Boltzmann measure (finite graph) : 1 � P dimer ( M ) = ν e , Z dimer e ∈ M where Z dimer is the dimer partition function.

Ising-dimer correspondence [Fisher] Ising model on toroidal graph G , coupling constants J . • Low temperature expansion on G ∗ → measure on polygonal contours of G . + − − −

Ising-dimer correspondence [Fisher] Ising model on toroidal graph G , coupling constants J . • Low temperature expansion on G ∗ → measure on polygonal contours of G . G G • G : Fisher graph of G .

Ising-dimer correspondence [Fisher] Ising model on toroidal graph G , coupling constants J . • Low temperature expansion on G ∗ → measure on polygonal contours of G . • G : Fisher graph of G . • Correspondence : Contour conf. → 2 | V ( G ) | dimer configurations. • Critical Ising model on G ↔ critical dimer model on G . � cot θ e original edges 2 ν e = 1 edges of the decorations .

Critical dimer model on infinite Fisher graph: periodic case • G : infinite Z 2 -periodic isoradial graph, G : corresponding Fisher graph. • Toroidal exhaustion {G n } = {G /n Z 2 } . G 1 : fundamental domain. Z n : partition function of G n , P n : Boltzmann measure of G n .

Critical dimer model on infinite Fisher graph: periodic case • G : infinite Z 2 -periodic isoradial graph, G : corresponding Fisher graph. • Toroidal exhaustion {G n } = {G /n Z 2 } . G 1 : fundamental domain. Z n : partition function of G n , P n : Boltzmann measure of G n . • Our goal is to: ◦ compute the free energy: 1 f = − lim n 2 log Z n . n →∞ ◦ obtain an explicit expression for a natural Gibbs measure: probability measure P such that if one fixes a perfect matching in an annular region, matchings inside and outside of the annulus are independent, and � P ( M ) α ν e , e ∈ M when M is a matching inside of the annulus (DLR conditions).

Kasteleyn matrix K of the graph G • Kasteleyn orientation of the graph: all elementary cycles are clockwise odd. • K : corresponding weighted oriented adjacency matrix,  ν uv if u ∼ v, u → v   K u,v = − ν uv if u ∼ v, u ← v  0 else .  • K 1 ( z, w ): is the Kasteleyn matrix of G 1 , with modified weights along edges crossing a dual horizontal and vertical cycle. w w w z 1/z z 1/z G 1 1/z z 1/w 1/w 1/w • det K 1 ( z, w ) is the dimer characteristic polynomial.

Free energy and Gibbs measure Theorem (Boutillier,dT) • The free energy of the dimer model on G is: 1 �� T 2 log(det K 1 ( z, w )) dz dw f = − 2(2 πi ) 2 z w • The weak limit of the Boltzmann measures P n defines a Gibbs measure P dimer on G . The probability of the subset of edges E = { e 1 = u 1 v 1 , · · · , e k = u k v k } being in a dimer configuration of G is given by: � k � � Pf(( K − 1 ) E ) , P ( e 1 , · · · , e k ) = K u i ,v i where i =1 Cof( K 1 ( z,w )) t K − 1 1 �� v,v ′ z x ′ − x w y ′ − y dz dw ( v,x,y )( v ′ ,x ′ ,y ′ ) = w . T 2 (2 πi ) 2 det K 1 ( z,w ) z [Probab. Theory Related Fields, 2010]

Idea of the proof (Gibbs measure) [Cohn, Kenyon, Propp; Kenyon, Okounkov, Sheffield] Theorem (Kasteleyn,Kenyon,...) The Boltzmann measure P n ( e 1 , · · · , e k ) is equal to: �� k � i =1 K u i ,v i − Pf( K 00 n ) E C + Pf( K 10 n ) E C + Pf( K 01 n ) E C + Pf( K 11 � � n ) E C 2 Z n n ) − 1 • Use Jacobi’s formula: Pf(( K θτ n ) E C ) = Pf( K θτ n )Pf(( K θτ E ). • Use Fourier to block diagonalize K θτ n . • Obtain Riemann sums. Show that they converge on a subsequence to the corresponding integral. • Use Sheffield’s theorem, which proves a priori existence of the limit.

Proof, continued • Sheffield’s theorem does not hold for non bipartite graphs. � 1 Pf( K 00 n ) EC • K 00 � n is never invertible ⇒ delicate estimate, = O . Z n n • To show convergence of the Riemann sums, need to know what are the zeros of det K 1 ( z, w ) on the torus T 2 .

Characteristic polynomial and the Laplacian • G : infinite Z 2 -periodic isoradial graph. • Laplacian on G , with weights tan( θ e ) is represented by the matrix ∆: � tan( θ uv ) if u ∼ v ∆ u,v = − � w ∼ u tan( θ uw ) if u = v. • ∆ 1 ( z, w ) : Laplacian matrix on G 1 with additional weights z and w . • Laplacian characteristic polynomial: det(∆ 1 ( z, w )). Theorem (Boutillier,dT) • There exists a constant c such that: det K 1 ( z, w ) = c det ∆ 1 ( z, w ) . • The curve { ( z, w ) ∈ C 2 : det K 1 ( z, w ) = 0 } is a Harnack curve and det K 1 ( z, w ) admits a unique double zero (1 , 1) on T 2 .

Local expression for K − 1 , general case • Discrete exponential function [Mercat,Kenyon] Exp : V ( G ) × V ( G ) × C → C , e i β k v u e i γ k Exp u , u ( λ ) = 1 Exp u , u k +1 ( λ ) = Exp u , u k ( λ )( λ + e iβ k )( λ + e iγ k ) ( λ − e iβ k )( λ − e iγ k ) .

Local expression for K − 1 Theorem (Boutillier,dT) The inverse of the Kasteleyn matrix K on G has the following local expression: 1 � K − 1 u,v = f u ( λ ) f v ( λ )Exp u , v log( λ ) dλ + c u,v , (2 πi ) 2 C uv where • C uv is a closed contour containing all poles of the integrand, and avoiding a half-line d uv . � 0 if u � = v • c u,v = ± 1 else . 4 [Comm. Math. Phys. 2010]

Proof (sketch) • Idea [Kenyon] • f v ( λ )Exp u , v ( λ ) is in the kernel of K . • Use singularities of the log: define contours of integrations in such a way that: � 0 if u � = v ( KK − 1 )( u, v ) = 1 if u = v.

Gibbs measure, local expression Theorem (Boutillier,dT) k � K ( u i , v i )Pf(( K − 1 ) E ) , P ( e 1 , · · · , e k ) = i =1 defines a Gibbs measure in dimer configurations of G . Proof. • [dT]: every finite, simply connected subgraph of a rhombus tiling can be completed by rhombi in order to become a periodic rhombus tiling of the plane. • Convergence of the Boltzmann measures in the periodic case. • Locality of the inverse Kasteleyn matrix. • Uniqueness of the inverse Kasteleyn matrix in the periodic case. • Kolmogorov’s extension theorem.

Consequences • Theorem (Baxter’s formula) Let G be a periodic isoradial graph. Then, the free energy of the critical Ising model on G is: f Ising = − log 2 1 θ e π log( θ e )+ 1 � π � �� � − L ( θ e ) + L 2 − θ e , 2 | V ( G 1 ) | π e ∈ E ( G 1 ) � θ where L ( θ ) = − 0 log(2 sin( t )) dt is Lobachevski’s function.   + − θ e  = 1 θ e • P Ising 4 − 2 π sin θ e .    J e • Spin/spin correlations are local. • Asymptotics computations for the dimer Gibbs measure.