Planar Ising model at criticality: state-of-the-art and perspectives Dmitry Chelkak, ´ ENS, Paris [ & PDMI, St. Petersburg (on leave) ] ICM, Rio de Janeiro, August 4, 2018
Planar Ising model at criticality: outline • Combinatorics Definition, phase transition Dimers and fermionic observables Spin correlations and fermions on double-covers Kadanoff–Ceva’s disorders and propagation equation Diagonal correlations and orthogonal polynomials • Conformal invariance at criticality S-holomorphic functions and Smirnov’s s-harmonicity Spin correlations: convergence to tau-functions More fields and CFT on the lattice Convergence of interfaces and loop ensembles Tightness of interfaces and ‘strong’ RSW • Beyond regular lattices: s-embeddings [2017+] [ two disorders inserted ] • Perspectives and open questions (c) Cl´ ement Hongler (EPFL)
Planar Ising model: definition [ Lenz, 1920 ] [ Centenary soon! ] • Lenz-Ising model on a planar graph G ∗ (dual to G ) is a random assignment of + / − spins to vertices of G ∗ (=faces of G ) according to � � � conf . σ ∈ {± 1 } V ( G ∗ ) � β � ∝ exp e = � uv � J uv σ u σ v P Z − 1 · � = e = � uv � : σ u � = σ v x uv , where J uv > 0 are interaction constants preassigned to edges � uv � , β = 1 / kT , and x uv = exp[ − 2 β J uv ]. • Remark: w/o magnetic field ⇒ ‘free fermion’. Ensemble of domain walls • Example: homogeneous model ( x uv = x ) on Z 2 . between ‘ + ’ and ‘ − ’ spins. ◦ Ising’25: no phase transition in 1D � doubts; ◦ Peierls’36: existence of the phase transition in 2(+)D; • ‘ + ’ boundary conditions √ ◦ Kramers-Wannier’41: x self-dual = 2 − 1; ⇒ collection of loops. ◦ Onsager’44: sharp phase transition at x crit = x self-dual . bullet ◦
√ 2 − 1 on Z 2 ] Planar Ising model: phase transition [ Kramers–Wannier’41: x crit = • Spin-spin correlations: • e.g., two spins at distance δ = 1 1 2 δ • 2n → ∞ along a diagonal. 2n x < x crit : does not vanish; ① = ① crit : power-law decay; x > x crit : exponential decay. Theorem [“diagonal correlations”, Kaufman–Onsager’49, Yang’52, McCoy–Wu’66+] : C ⋄ [ σ 0 σ 2 ♥ ] = ( 1 − tan 4 θ ) 1 / 4 > 0 . (i) For ① = tan 1 2 θ < x crit , one has lim ♥ →∞ E ① � 2 � n · � n − 1 � � k − n ∼ C 2 σ · ( 2 ♥ ) − 1 (ii) At criticality, E ① = ① crit 1 4 . [ σ 0 σ 2 ♥ ] = 1 − C ⋄ π k =1 4 k 2 Remark: Many highly nontrivial results on the spin correlations in the infinite volume are known. Reference: B.M.McCoy – T.T.Wu “The two-dimensional Ising model”.
√ 2 − 1 on Z 2 ] Planar Ising model: phase transition [ Kramers–Wannier’41: x crit = • Spin-spin correlations: • e.g., two spins at distance δ = 1 1 2 δ • 2n → ∞ along a diagonal. 2n x < x crit : does not vanish; ① = ① crit : power-law decay; x > x crit : exponential decay. • Domain walls structure: x < x crit : “straight”; ① = ① crit : SLE (3), CLE (3); ① > ① crit : SLE (6), CLE (6). [ this is not proved ] x < x crit ① = ① crit ① > ① crit
Combinatorics: planar Ising model via dimers (’60s) and fermionic observables 1 1 2 # V ( G ) -to-1 1 X e X e 1 ← − − − − − − − − − → 1 1 1 Fisher’s graph ● F : vertices are corners and oriented edges of G . , Z ∼ • Kasteleyn’s theory: F = F = − F ⊤ = Pf[ F ] • Fermions: � φ ❝ φ ❞ � := F − 1 ( ❝ , ❞ ) = −� φ ❞ φ ❝ � • Pfaffian (or Grassmann variables) formalism: � φ c 1 . . . φ c 2 k � = Pf [ � φ c p φ c q � ] 2 k p , q =1
Combinatorics: planar Ising model via dimers (’60s) and fermionic observables There are other 1 combinatorial 1 1 1 -1 correspondences ±1 1 X e X e 1 ±1 1 of the same kind: 1 1 1 1 1 ∼ Z Pf[ F ] = ∼ Fisher’s graph ● F : vertices are = Pf[ K ] Kasteleyn’s terminal graph ● K , ∼ Pf[ C ] = corners and oriented edges of G . vertices = oriented edges of G . • Kasteleyn’s theory: F = F = − F ⊤ • Fermions: � φ ❝ φ ❞ � := F − 1 ( ❝ , ❞ ) = −� φ ❞ φ ❝ � 1 1 ± 1 1 • Pfaffian (or Grassmann variables) formalism: 1 -1 X e � φ c 1 . . . φ c 2 k � = Pf [ � φ c p φ c q � ] 2 k p , q =1 ● C : vertices = corners of G .
Combinatorics: planar Ising model via dimers (’60s) and fermionic observables There are other 1 combinatorial 1 1 1 -1 correspondences ±1 1 X e X e 1 ±1 1 of the same kind: 1 1 1 1 1 ∼ Z Pf[ F ] = ∼ Fisher’s graph ● F : vertices are = Pf[ K ] Kasteleyn’s terminal graph ● K , ∼ Pf[ C ] = corners and oriented edges of G . vertices = oriented edges of G . • Two other useful techniques: Reference: arXiv:1507.08242 •• Kac–Ward matrix is equivalent to K; (w/ D. Cimasoni and A. Kassel) •• Smirnov’s fermionic observables (2000s) are “Revisiting the combinatorics •• combinatorial expansions of Pf[ F V ( ● F ) \{ ❝ , ❞ } ] . of the 2D Ising model”
Combinatorics: spin correlations and fermions on double-covers u 1 Observation: 1 1 1 1 1 X e 1 – X e 1 E [ σ ✉ 1 ...σ ✉ ♥ ] 1 1 1 1 = Pf [ F [ ✉ 1 ,.., ✉ ♥ ] ] 1 1 1 u 2 Pf [ F ] Fisher’s graph ● F : vertices are One changes ① ❡ �→ − ① ❡ along corners and oriented edges of G . γ [ ✉ 1 , ✉ 2 ] to compute E [ σ ✉ 1 σ ✉ 2 ] .
Combinatorics: spin correlations and fermions on double-covers u 1 Observation: 1 1 1 1 1 X e 1 – X e 1 E [ σ ✉ 1 ...σ ✉ ♥ ] 1 1 1 1 = Pf [ F [ ✉ 1 ,.., ✉ ♥ ] ] 1 1 1 u 2 Pf [ F ] Fisher’s graph ● F : vertices are One changes ① ❡ �→ − ① ❡ along corners and oriented edges of G . γ [ ✉ 1 , ✉ 2 ] to compute E [ σ ✉ 1 σ ✉ 2 ] . Corollary: Let w 1 ∼ u 1 . The ratio E [ σ ✇ 1 σ ✉ 2 ...σ ✉ ♥ ] E [ σ ✉ 1 σ ✉ 2 ...σ ✉ ♥ ] can be expressed via F − 1 [ ✉ 1 ,.., ✉ ♥ ] . Remark: Instead of fixing cuts one can view ❋ − 1 [ ✉ 1 ,.., ✉ ♥ ] ( ❝ ♭ , ❞ ) = − ❋ − 1 [ ✉ 1 ,.., ✉ ♥ ] ( ❝ ♯ , ❞ ) as [ ✉ 1 ,.., ✉ ♥ ] of the graph G F ramified over faces u 1 , .., u n . a spinor on the double-cover ● F
Combinatorics: Kadanoff–Ceva(’71) disorders and propagation equation • Given (an even number of) vertices v 1 , ..., v m , • consider the Ising model on (the faces of) the • double-cover G [ v 1 ,.., v m ] ramified over v 1 , ..., v m • with the spin-flip symmetry constraint σ u ♭ = − σ u ♯ • provided that u ♭ , u ♯ lie over the same face u of G . • Define � µ ✈ 1 ...µ ✈ ♠ σ ✉ 1 ...σ ✉ ♥ � := E [ ✈ 1 ,.., ✈ ♠ ] [ σ ✉ 1 ...σ ✉ ♥ ] · Z [ ✈ 1 ,.., ✈ ♠ ] / Z . • [!] By definition, this (formal) correlator changes • the sign when one of u k goes around of one of v s . [ two disorders inserted ] (c) Cl´ ement Hongler (EPFL)
Combinatorics: Kadanoff–Ceva(’71) disorders and propagation equation • Given (an even number of) vertices v 1 , ..., v m , • consider the Ising model on (the faces of) the • double-cover G [ v 1 ,.., v m ] ramified over v 1 , ..., v m • with the spin-flip symmetry constraint σ u ♭ = − σ u ♯ • provided that u ♭ , u ♯ lie over the same face u of G . • Define � µ ✈ 1 ...µ ✈ ♠ σ ✉ 1 ...σ ✉ ♥ � := E [ ✈ 1 ,.., ✈ ♠ ] [ σ ✉ 1 ...σ ✉ ♥ ] · Z [ ✈ 1 ,.., ✈ ♠ ] / Z . • For a corner c of G , define χ ❝ := µ ✈ ( ❝ ) σ ✉ ( ❝ ) . • Proposition: If all vertices v ( c k ) are distinct, then ±� χ ❝ 1 ...χ ❝ 2 ❦ � = ±� φ ❝ 1 ...φ ❝ 2 ❦ � . [ two disorders inserted ] • Proof: expand both sides combinatorially on G . (c) Cl´ ement Hongler (EPFL)
Combinatorics: Kadanoff–Ceva(’71) disorders and propagation equation u 1 Parameterization: v 0 v 1 c 00 c 10 z ① ❡ = tan 1 c 01 2 θ ❡ u 0 • Propagation equation: Let X ( c ) := � χ c O [ µ, σ ] � . • Then ❳ ( ❝ 00 ) = ❳ ( ❝ 01 ) cos θ ❡ + ❳ ( ❝ 10 ) sin θ ❡ . • For a corner c of G , define χ ❝ := µ ✈ ( ❝ ) σ ✉ ( ❝ ) . • Proposition: If all vertices v ( c k ) are distinct, then ±� χ ❝ 1 ...χ ❝ 2 ❦ � = ±� φ ❝ 1 ...φ ❝ 2 ❦ � . [ two disorders inserted ] • Proof: expand both sides combinatorially on G . (c) Cl´ ement Hongler (EPFL)
Combinatorics: Kadanoff–Ceva(’71) disorders and propagation equation Parameterization: 1 sin( ϑ e ) sin( ϑ e ) ① ❡ = tan 1 2 θ ❡ cos( ϑ e ) cos( ϑ e ) cos( ϑ e ) sin( ϑ e ) ● D : bipartite (Wu–Lin’75). • Propagation equation: Let X ( c ) := � χ c O [ µ, σ ] � . Fact: D − 1 = C − 1 + local . • Then ❳ ( ❝ 00 ) = ❳ ( ❝ 01 ) cos θ ❡ + ❳ ( ❝ 10 ) sin θ ❡ . • [ Perk’80, Dotsenko–Dotsenko’83, . . . , Mercat’01 ] 1 • Bosonization: To obtain a combinatorial 1 ± 1 1 1 • representation of the model via dimers on G D -1 X e • one should start with two Ising configurations ● C : vertices = corners of G . • [ e.g., see Dub´ edat’11, Boutillier–de Tili` ere’14 ]
Recommend
More recommend
