Universality and conformal invariance in the 2D critical Ising model - PowerPoint PPT Presentation

Universality and conformal invariance in the 2D critical Ising model Dmitry Chelkak (St.Petersburg) joint work with Stanislav Smirnov (Geneva) Stochastic Processes and Their Applications � 2009 Special Session �SLE� Berlin, July 29

Critical Ising model on the square grid: [S. Smirnov. Towards conformal invariance of 2D lattice models. Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22�30, 2006.] spin-Ising model FK-Ising model Interface → SLE 3 Interface → SLE 16 / 3 as mesh → 0 . as mesh → 0 .

Main steps: I. �Combinatorics� : Construction of the martingale observable (Ω δ ; a δ , b δ ) ( z δ ) , z δ ∈ Ω δ , solving some (�holomorphic fermion�) F δ discrete boundary value problem such that ◮ F δ is discrete holomorphic (w.r.t. z δ ) for all (Ω δ ; a δ , b δ ) ; (Ω δ \ γ δ [0 , n ]; γ δ ( n ) , b δ ) is a martingale (for any �xed z δ ) ◮ F δ w.r.t. the (discrete) interface γ δ growing from a δ .

Main steps: I. �Combinatorics� : Construction of the martingale observable (Ω δ ; a δ , b δ ) ( z δ ) , z δ ∈ Ω δ , solving some (�holomorphic fermion�) F δ discrete boundary value problem such that ◮ F δ is discrete holomorphic (w.r.t. z δ ) for all (Ω δ ; a δ , b δ ) ; (Ω δ \ γ δ [0 , n ]; γ δ ( n ) , b δ ) is a martingale (for any �xed z δ ) ◮ F δ w.r.t. the (discrete) interface γ δ growing from a δ . II. �Complex analysis� : F δ is uniformly close (w.r.t. all possible simply-connected domains, including those with rough boundaries) to its continuous (conformally covariant) counterpart f (Ω δ ; a δ , b δ ) [solving the continuous version of the same boundary value problem]

Main steps: I. �Combinatorics� : Construction of the martingale observable (Ω δ ; a δ , b δ ) ( z δ ) , z δ ∈ Ω δ , solving some (�holomorphic fermion�) F δ discrete boundary value problem such that ◮ F δ is discrete holomorphic (w.r.t. z δ ) for all (Ω δ ; a δ , b δ ) ; (Ω δ \ γ δ [0 , n ]; γ δ ( n ) , b δ ) is a martingale (for any �xed z δ ) ◮ F δ w.r.t. the (discrete) interface γ δ growing from a δ . II. �Complex analysis� : F δ is uniformly close (w.r.t. all possible simply-connected domains, including those with rough boundaries) to its continuous (conformally covariant) counterpart f (Ω δ ; a δ , b δ ) [solving the continuous version of the same boundary value problem] III. �Probability� : ⇒ discrete interfaces converge to SLE ( κ ) , where f ( C + \ SLE κ [0 , t ]; SLE κ ( t ) , ∞ ) ( z ) is a martingale for all z ∈ C + . κ :

More general lattices. Y − ∆ invariance. ↔ AB + C BC + A = ab bc = CA + B = ABC + 1 ca 1

More general lattices. Y − ∆ invariance. ↔ & ↔ ↔ & AB + C BC + A = '06] Local weights [R. Costa-Santos ab bc satisfying relation Y − ∆ naturally lead to the isoradial = CA + B = ABC + 1 embedding of the graph. ca 1 Isoradial embedding means that all faces can be inscribed into circles of equal radii δ (the mesh of the �lattice�).

Isoradial graphs. Notations. ◮ isoradial graph Γ (black vertices), ◮ dual isoradial graph Γ ∗ (gray vertices); ◮ rhombic lattice ( Λ = Γ ∪ Γ ∗ , blue edges) ◮ and the set ♦ = Λ ∗ (white �diamonds�). ( ♠ ): we assume that rhombi angles are uniformly bounded away from 0 and π .

Critical Ising model on isoradial graphs. [C. Mercat '01; V. Riva, J. Cardy '06; C. Boutillier, B. de Tili� ere '09; ...] tan θ ij � � Z = 2 config . w i � = w j Observable (discrete holomorphic martingale): F δ ( z ) := Z config . : a � z · e − i 2 winding ( a � z ) z ∈ ♦ . 2 winding ( a � b ) , Z config . : a � b · e − i

Riemann-Hilbert boundary value problem. ◮ F ( z ) is holomorphic in Ω ; 1 2 ] = 0 on the boundary ∂ Ω \ { a } ; ◮ Im [ F ( ζ )( τ ( ζ )) ◮ proper normalization at b : 1 2 = +1 ; ◮ τ ( b ) ◮ ∂ H b = F 2 ( b ) = 1 ; � � ∂ y ◮ H is nonnegative everywhere in Ω . Remark. F is well de�ned in rough domains via H = Im F 2 dz � which is the imaginary part of the conformal mapping from (Ω; a , b ) onto the upper half-plane ( C + ; ∞ , 0) normalized at b .

Discrete complex analysis on isoradial graphs. [R.J. Du�n '60s; C. Mercat '01; R. Kenyon '02; A. Bobenko, C. Mercat, Yu. Suris '05 ...] Di�erence operators ∆ δ , ∂ δ , ∂ δ : H : Λ → C ; ∂ δ H ( z s ) := 1 � H ( u s ) − H ( u ) + H ( w s +1 ) − H ( w s ) � ; 2 u s − u w s +1 − w s ∂ δ H ( z s ) := 1 � H ( u s ) − H ( u ) + H ( w s +1 ) − H ( w s ) � ; 2 u s − u w s +1 − w s

Discrete complex analysis on isoradial graphs. [R.J. Du�n '60s; C. Mercat '01; R. Kenyon '02; A. Bobenko, C. Mercat, Yu. Suris '05 ...] Di�erence operators ∆ δ , ∂ δ , ∂ δ : H : Λ → C ; F : ♦ → C . ∂ δ F ( u ) (= ( ∂ δ ) ∗ F ( u )) := i � − ( w s +1 − w s ) F ( z s ); 2 µ δ Γ ( u ) z s ∼ u ∂ δ H ( z s ) := 1 � H ( u s ) − H ( u ) + H ( w s +1 ) − H ( w s ) � ; 2 u s − u w s +1 − w s ∂ δ H ( z s ) := 1 � H ( u s ) − H ( u ) + H ( w s +1 ) − H ( w s ) � ; 2 u s − u w s +1 − w s

Discrete complex analysis on isoradial graphs. [R.J. Du�n '60s; C. Mercat '01; R. Kenyon '02; A. Bobenko, C. Mercat, Yu. Suris '05 ...] Di�erence operators ∆ δ , ∂ δ , ∂ δ : H : Λ → C ; F : ♦ → C . ∂ δ F ( u ) (= ( ∂ δ ) ∗ F ( u )) := i � − ( w s +1 − w s ) F ( z s ); 2 µ δ Γ ( u ) z s ∼ u ∂ δ H ( z s ) := 1 � H ( u s ) − H ( u ) + H ( w s +1 ) − H ( w s ) � ; 2 u s − u w s +1 − w s

Discrete complex analysis on isoradial graphs. [R.J. Du�n '60s; C. Mercat '01; R. Kenyon '02; A. Bobenko, C. Mercat, Yu. Suris '05 ...] Di�erence operators ∆ δ , ∂ δ , ∂ δ : H : Λ → C ; F : ♦ → C . ∂ δ F ( u ) (= ( ∂ δ ) ∗ F ( u )) := i � − ( w s +1 − w s ) F ( z s ); 2 µ δ Γ ( u ) z s ∼ u ∂ δ H ( z s ) := 1 � H ( u s ) − H ( u ) + H ( w s +1 ) − H ( w s ) � ; 2 u s − u w s +1 − w s 1 ∆ δ H ( u ) := 4 ∂ δ ∂ δ H ( u ) = � tan θ s · [ H ( u s ) − H ( u )] . µ δ Γ ( u ) u s ∼ u

Discrete complex analysis on isoradial graphs. Corresponding random walk on Γ : RW ( t +1) = RW ( t ) + ξ ( t ) RW ( t ) , where ξ ( t ) are independent and tan θ k P ( ξ u = u k − u ) = . � n s =1 tan θ s Then: E [ Re ξ u ] = E [ Im ξ u ] = 0 , E [( Re ξ u ) 2 ] = E [( Im ξ u ) 2 ] = δ 2 · T u E [ Re ξ u Im ξ u ] = 0 , (where T u = P n ‹P n s =1 tan θ s ). s =1 sin 2 θ s

Discrete complex analysis on isoradial graphs. Convergence for discrete harmonic functions: ◮ The uniform (w.r.t. (a) shape of the simply-connected domain Γ and (b) structure of the underlying isoradial graph) Ω δ C 1 -convergence in the bulk of the basic objects of the discrete potential theory to their continuous counterparts holds true. (i) harmonic measure (exit probability) ω δ ( · ; a δ b δ ; Ω δ Γ ) of boundary arcs a δ b δ ⊂ ∂ Ω δ Γ ; Γ ( · ; v δ ) , v δ ∈ Int Ω δ (ii) Green function G δ Γ ; Ω δ Γ ) = ω δ ( · ; { a δ } ; Ω δ Γ ) Γ ) , a δ ∈ ∂ Ω δ (iii) Poisson kernel P δ ( · ; v δ ; a δ ; Ω δ Γ ω δ ( v δ ; { a δ } ; Ω δ normalized at the inner point v δ ∈ Int Ω δ Γ ; Γ ) , a δ , o δ ∈ ∂ Ω δ (iv) Poisson kernel P δ Γ , normalized at the o δ ( · ; a δ ; Ω δ boundary by the discrete analogue of the condition ∂ n P | o δ = − 1 . ∂

Discrete complex analysis on isoradial graphs. S-holomorphic functions: We call F (de�ned on some subset of ♦ ) s-holomorphic , if Pr[ F ( z 1 ) ; [ i ( w − u )] − 1 2 ] = Pr[ F ( z 2 ) ; [ i ( w − u )] − 1 2 ] for any two neighbors z 0 ∼ z 1 . ◮ implies standard discrete holomorphicity (i.e., ∂ δ F = 0 ); ◮ holds for observables in the critical Ising model ; ◮ can be reformulated as �propagation equation� (or Dotsenko-Dotsenko equation) for some discrete spinor de�ned on the (double covering of) edges uw [cf. C.Mercat '01]

Discrete complex analysis on isoradial graphs. Convergence for the �spin-Ising observable�: (A) S-holomorphicity: F δ ( z ) is s-holomorphic inside Ω δ ♦ . 1 (B) Boundary conditions: Im [ F δ ( ζ )( τ ( ζ )) 2 ] = 0 for all ζ ∈ ∂ Ω δ ♦ except a δ , where τ ( ζ ) is the tangent vector at ζ oriented in the 1 counterclockwise direction (and τ ( b δ ) 2 = +1 ). (C) Normalization at the target point: F δ ( b δ ) = 1 . Theorem (Ch.-Smirnov): After some re-normalization by constants K δ ≍ 1 (which depend on the structure of ♦ δ but don't depend on the shape of Ω δ ), the solution of the discrete boundary value problem (A)&(B)&(C) is uniformly close in the bulk to its continuous counterpart f (Ω δ ; a δ , b δ ) .

Namely, there exists ε ( δ ) = ε ( δ, r , R , s , t ) such that for all simply-connected discrete domains (Ω δ ♦ ; a δ , b δ ) having �straight� boundary near b δ and z δ ∈ Ω δ ♦ the following holds true: if B ( z δ , r ) ⊂ Ω δ ⊂ B ( z δ , R ) , then | K δ · F δ ( z δ ) − f (Ω δ ; a δ , b δ ) ( z δ ) | � ε ( δ ) → 0 as δ → 0 (uniformly w.r.t. the shape of Ω δ and the structure of ♦ δ ). Technical remark: we assume that discrete domains Ω δ contain some �xed rectangle [ − s , s ] × [0 , t ] and their boundaries near target points b δ ≈ 0 approximate the straight segment [ − s , s ] ;