Crossing probabilities in the critical 2D Ising model Dmitry Chelkak - PowerPoint PPT Presentation

Crossing probabilities in the critical 2D Ising model Dmitry Chelkak (PDMI Steklov, St.Petersburg) joint work with Stanislav Smirnov (Geneva) arXiv:0910.2045 : �Universality in the 2D Ising model and conformal invariance of fermionic observables� , 50pp. Conformal Structures and Dynamics (CODY) Seillac, France, May 2�8, 2010

2D Ising model: (square grid) Spins σ i = +1 or − 1 . Hamiltonian: H = − � � ij � σ i σ j . Partition function: P ( conf . ) ∼ e − β H ∼ x # � + −� , where x = e − 2 β ∈ [0 , 1] .

2D Ising model: (square grid) Spins σ i = +1 or − 1 . Hamiltonian: H = − � � ij � σ i σ j . Partition function: P ( conf . ) ∼ e − β H ∼ x # � + −� , where x = e − 2 β ∈ [0 , 1] . Other �lattices� (planar graphs): � ij � J ij σ i σ j . H = − � P ( conf . ) ∼ � � ij � : σ i � = σ j x ij , x ij ∈ [0 , 1] .

Phase transition, criticality: x > x crit x = x crit x < x crit (Dobrushin boundary values: two marked points a , b on the boundary; +1 on the arc ( ab ) , − 1 on the opposite arc ( ba ) ) [Peierls `36; Kramers-Wannier '41]: x crit = 1 √ 2+1

Conformal invariance: Quantities (spin correlations, crossing probabilities, etc.) [Cardy's formula for percolation, etc.] � Geometry (interfaces, loop ensembles, etc.) [Schramm's SLEs, CLEs, etc.]

Conformal invariance: Quantities (spin correlations, crossing probabilities, etc.) [Cardy's formula for percolation, etc.] � Geometry (interfaces, loop ensembles, etc.) [Schramm's SLEs, CLEs, etc.] � ⇑ �: SLE computations

Conformal invariance: Quantities (spin correlations, crossing probabilities, etc.) [Cardy's formula for percolation, etc.] � Geometry (interfaces, loop ensembles, etc.) [Schramm's SLEs, CLEs, etc.] � ⇑ �: SLE computations � ⇓ �: Conformal martingale principle Ref: S. Smirnov. Towards conformal invariance of 2D lattice models. [ Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22�30, 2006 ]

Spin- and FK-Ising models (random cluster representation): P ( spins conf . ) ∼ x # � + −� � � � = x + (1 − x ) · χ s ( i )= s ( j ) < ij >

Spin- and FK-Ising models (random cluster representation): P ( spins conf . ) ∼ x # � + −� � � � = x + (1 − x ) · χ s ( i )= s ( j ) < ij > � x ) # open x # closed = (1 − edges conf .

Spin- and FK-Ising models (random cluster representation): P ( spins & edges conf . ) ∼ (1 − x ) # open x # closed Open edges connect equal spins (but not all) Erase spins:

Spin- and FK-Ising models (random cluster representation): P ( spins & edges conf . ) ∼ (1 − x ) # open x # closed Open edges connect equal spins (but not all) Erase spins: P ( edges conf . ) ∼ 2 # clusters (1 − x ) # open x # closed

Spin- and FK-Ising models (random cluster representation): P ( edges conf . ) ∼ 2 # clusters (1 − x ) # open x # closed ∼ 2 # clusters [(1 − x ) / x ] # open

Spin- and FK-Ising models (random cluster representation): P ( edges conf . ) ∼ 2 # clusters (1 − x ) # open x # closed ∼ 2 # clusters [(1 − x ) / x ] # open ∼ √ √ # loops [(1 − x ) / ( x 2)] # open 2 since # loops − # open edges = 2# clusters + const

Spin- and FK-Ising models (random cluster representation): Self-dual case ( x = x crit ): √ # loops P ( loops conf . ) ∼ 2

Spin- and FK-Ising models (random cluster representation): Self-dual case ( x = x crit ): √ # loops P ( loops conf . ) ∼ 2 Then P spin ( s ( i )= s ( j )) = 1 2(1 + P FK ( i ↔ j ))

Convergence to SLE. Square lattice (Smirnov): Spin-Ising Theorem: FK-Ising Theorem: Interface → SLE(3) Interface → SLE(16/3)

Universality. Isoradial graphs/rhombic lattices: Spin-Ising Theorem: FK-Ising Theorem: Interface → SLE(3) Interface → SLE(16/3) √ tan θ ( z ) sin θ ( z ) # loops � � � � Z = Z = 2 2 2 z : ⊕↔⊖ config . z config .

Universality. Isoradial graphs/rhombic lattices: FK-Ising Theorem: Interface → SLE(16/3) FK-Ising local weights: sin θ 2 = sin( π 4 − θ 2 ) √ sin θ ( z ) # loops � � Z = 2 2 config . z

Universality. Isoradial graphs/rhombic lattices: sin θ FK-Ising local weights: 2 = 2 ) =: r ( θ ) sin( π 4 − θ satis�es r (0) = 0 and Y − ∆ invariance: if α + β + γ = π 2 , then √ 2 · r ( α ) r ( β ) r ( γ ) . 1 = r ( α ) r ( β ) + r ( α ) r ( γ ) + r ( β ) r ( γ ) + ↔

Universality. Isoradial graphs/rhombic lattices: ↔ Y- ∆ invariance :

Universality. Isoradial graphs/rhombic lattices: ↔ Y- ∆ invariance : ↔ √ 2 · r ( α ) r ( β ) r ( γ ) 1 = r ( α ) r ( β ) + r ( α ) r ( γ ) + r ( β ) r ( γ ) +

Conformal martingale (discrete fermionic observable): FK-Ising Theorem: Interface → SLE(16/3) √ sin θ ( z ) # loops � � Z = 2 2 config . z

Conformal martingale (discrete fermionic observable): Discrete holomorphic FK-Ising Theorem: observable having the Interface → SLE(16/3) martingale property: F δ = E χ [ z ∈ γ ] · e − i 2 · wind ( γ, b → z ) , where z ∈ ♦ . √ sin θ ( z ) # loops � � Z = 2 2 z config .

Conformal martingale (discrete fermionic observable): Discrete holomorphic Boundary Value Problem: observable having the ◮ F ( z ) is holomorphic in Ω ; martingale property: 1 2 ] = 0 ◮ Im [ F ( ζ )( τ ( ζ )) for ζ ∈ ∂ Ω \ { a , b } , F δ = E χ [ z ∈ γ ] · e − i 2 · wind ( γ, b → z ) , where τ ( ζ ) goes from a to b ; where z ∈ ♦ . ◮ (mult.) normalization.

Conformal martingale (discrete fermionic observable): Discrete holomorphic Boundary Value Problem: observable having the ◮ F ( z ) is holomorphic in Ω ; martingale property: 1 2 ] = 0 ◮ Im [ F ( ζ )( τ ( ζ )) for ζ ∈ ∂ Ω \ { a , b } , F δ = E χ [ z ∈ γ ] · e − i 2 · wind ( γ, b → z ) , where τ ( ζ ) goes from a to b ; where z ∈ ♦ . ◮ (mult.) normalization. Solution: Φ ′ ( z ) , � F ( z ) = Φ : (Ω; a , b ) → ( S , −∞ , + ∞ ) , S = R × (0 , 1) .

Universality. Convergence to SLE (FK-Ising): F δ is a discrete holomorphic martingale. Then: ◮ Take a �discrete integral� H δ := Im ( F δ ) 2 ( z ) d δ z � (miraculously, it is well de�ned); ◮ H δ is NOT discrete harmonic, so prove that it is �approximately� harmonic;

Universality. Convergence to SLE (FK-Ising): F δ is a discrete holomorphic martingale. Then: ◮ Take a �discrete integral� H δ := Im ( F δ ) 2 ( z ) d δ z � (miraculously, it is well de�ned); ◮ H δ is NOT discrete harmonic, so prove that it is �approximately� harmonic; ◮ Prove that H δ is uniformly (w.r.t. Ω ) close to its eventual limit Im Φ = ω ( · , ba , Ω) inside Ω ; √ ◮ Prove that F δ is uniformly close to Φ ′ inside Ω ; This needs some work (see arXiv:0910.2045,0810.2188 ).

Universality. Convergence to SLE (FK-Ising): F δ is a discrete holomorphic martingale. Then: ◮ Take a �discrete integral� H δ := Im ( F δ ) 2 ( z ) d δ z � (miraculously, it is well de�ned); ◮ H δ is NOT discrete harmonic, so prove that it is �approximately� harmonic; ◮ Prove that H δ is uniformly (w.r.t. Ω ) close to its eventual limit Im Φ = ω ( · , ba , Ω) inside Ω ; √ ◮ Prove that F δ is uniformly close to Φ ′ inside Ω ; This needs some work (see arXiv:0910.2045,0810.2188 ). ◮ Then deduce convergence of an interface to SLE (16 / 3) from the convergence of the martingale observable.

Universality. Convergence to SLE (FK-Ising): ◮ Then deduce convergence of an interface to SLE (16 / 3) from the convergence of the martingale observable. Interfaces → SLE(16/3) . In which topology? ◮ Convergence of driving forces in the Loewner equation. Directly follows from the convergence of observable. ◮ Convergence of curves themselves. Needs some a priori information (estimates of some crossing probabilities). (Aizenman, Burchard, '99; Kemppainen, Smirnov '09)

FK-Ising crossing probability: vs. P δ Q δ

FK-Ising crossing probability: P δ Theorem : For all r , R , t > 0 there exists ε ( δ ) → 0 as δ → 0 such that if B (0 , r ) ⊂ Ω δ ⊂ B (0 , R ) and either both ω (0; Ω δ ; a δ b δ ) , ω (0; Ω δ ; c δ d δ ) or both ω (0; Ω δ ; b δ c δ ) , ω (0; Ω δ ; d δ a δ ) are � t (i.e., quadrilateral Ω δ has no neighboring small arcs), then Q δ | P δ − P (Ω δ ; a δ , b δ , c δ , d δ ) | � ε ( δ ) (uniformly w.r.t. Ω δ and ♦ δ ), where P depends only on the conformal modulus of (Ω δ ; a δ , b δ , c δ , d δ ) .

FK-Ising crossing probability: P δ In the half-plane H : for u ∈ [0 , 1] , P ( H ; [1 − u , 1] ↔ [ ∞ , 0]) 1 − √ 1 − u � = . 1 −√ u + 1 −√ 1 − u � � Q δ This is a special case of a hypergeometric formula for crossings in a general FK model. In the Ising case it becomes algebraic and furthermore can be rewritten in several ways.