Hypergeometric SLE and Convergence of Multiple Interfaces in Lattice - PowerPoint PPT Presentation

Hypergeometric SLE and Convergence of Multiple Interfaces in Lattice Models Hao Wu Yau Mathematical Sciences Center, Tsinghua University, China Hao Wu (THU) Hypergeometric SLE 1 / 32

Background Table of contents Background 1 Ising model SLE Hypergeometric SLE 2 More complicated b.c. 3 Pure Partition Functions 4 Hao Wu (THU) Hypergeometric SLE 3 / 32

Background Ising model Ising Model Curie temperature [Pierre Curie, 1895] Ferromagnet exhibits a phase transition by losing its magnetization when heated above a critical temperature. Ising Model [Lenz, 1920] A model for ferromagnet, to understand the critical temperature G = ( V , E ) is a finite graph ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ σ ∈ {⊕ , ⊖} V ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ The Hamiltonian a b ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ � H ( σ ) = − σ x σ y ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ x ∼ y ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ Hao Wu (THU) Hypergeometric SLE 4 / 32

Background Ising model Ising Model ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ Ising model is the probability ⊕ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ measure of inverse temperature ⊖ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ β > 0 : ⊖ ⊖ ⊖ ⊖ a b ⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊕ µ β, G [ σ ] ∝ exp( − β H ( σ )) ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ Kramers-Wannier, Onsager-Kaufman, 1940 Ising model on Z 2 : √ β c = 1 2 log( 1 + 2 ) . Interface Conformal invariance + Domain Markov Property Hao Wu (THU) Hypergeometric SLE 5 / 32

Background SLE SLE (Schramm Loewner Evolution) Random fractal curves in D ⊂ C from a to b . Candidates for the scaling limit of discrete Statistical Physics models. b D ϕ ( D ) Conformal invariance : γ ϕ ( b ) If γ is in D from a to b , ϕ ( γ ) ϕ and ϕ : D → ϕ ( D ) conformal map, then ϕ ( γ ) d ∼ the one in ϕ ( D ) from ϕ ( a ) to ϕ ( b ) . a ϕ ( a ) D Domain Markov Property : γ [ t, ∞ ) the conditional law of γ [0 , t ] a b γ [ t , ∞ ) given γ [ 0 , t ] d ∼ the one in D \ γ [ 0 , t ] from γ ( t ) to b . γ ( t ) Hao Wu (THU) Hypergeometric SLE 6 / 32

Background SLE Examples of SLE Lemma [Schramm 1999] There exists a one-parameter family of random curves that satisfies Conformal Invariance and Domain Markov Property : SLE κ for κ ≥ 0. Simple, κ ∈ [ 0 , 4 ] ; Self-touching, κ ∈ ( 4 , 8 ) ; Space-filling, κ ≥ 8. κ = 2 : LERW κ = 8 : UST (Lawler, Schramm, Werner) κ = 3 : Critical Ising κ = 16 / 3 : FK-Ising (Chelkak, Duminil-Copin, Hongler, Kemppainen, Smirnov) κ = 6 : Percolation Courtesy to Tom Kennedy. (Camia, Newman, Smirnov) Hao Wu (THU) Hypergeometric SLE 7 / 32

Background SLE Critical Ising Thm [Chelkak, Duminil-Copin, Hongler, Kemppainen, Smirnov 2010] The interface of critical Ising model on Z 2 with Dobrushin boundary condition converges to SLE ( 3 ) . Their Strategy Tightness : RSW Identify the scaling limit : Holomorphic observable Hao Wu (THU) Hypergeometric SLE 8 / 32

Background SLE Other results on the convergence? Thm [Chelkak, Duminil-Copin, Hongler, Kemppainen, Smirnov 2010] The interface of critical Ising model on Z 2 with Dobrushin boundary condition converges to SLE ( 3 ) . Different Models? Many conjectures. Different lattices? Universality : open. Different Boundary Conditions? Some results. Hao Wu (THU) Hypergeometric SLE 9 / 32

Background SLE Open Question : Other Models Conjecture For q ≤ 4, the interface of critical Random Cluster Model converges to SLE ( κ ) where κ = 4 π/ arccos ( −√ q / 2 ) . Conjecture The interface of Double Dimer Model converges to SLE ( 4 ) . Hao Wu (THU) Hypergeometric SLE 10 / 32

Background SLE Open Question : Universality Thm [Smirnov 2000] The interface of critical site percolation on triangular lattice converges to SLE ( 6 ) . Conjecture The interface of critical bond percolation on square lattice converges to SLE ( 6 ) . Hao Wu (THU) Hypergeometric SLE 11 / 32

Background SLE Other results on the convergence? Thm [Chelkak, Duminil-Copin, Hongler, Kemppainen, Smirnov 2010] The interface of critical Ising model on Z 2 with Dobrushin boundary condition converges to SLE ( 3 ) . Different Models? Many conjectures. Different lattices? Universality : open. Different Boundary Conditions? Some results. Hao Wu (THU) Hypergeometric SLE 12 / 32

Hypergeometric SLE Table of contents Background 1 Ising model SLE Hypergeometric SLE 2 More complicated b.c. 3 Pure Partition Functions 4 Hao Wu (THU) Hypergeometric SLE 13 / 32

Hypergeometric SLE Critical Ising in Quad Thm [Izyurov 2014, W. 2017] The interface of critical Ising model on Z 2 with alternating boundary condition converges to Hypergeometric SLE 3 , denoted by hSLE 3 . Q1 : What is Hypergeometric SLE? Q2 : Why are they the limit? Q3 : How do we prove the convergence? Answer to Q1 : random fractal curves in quad q = (Ω; x 1 , x 2 , x 3 , x 4 ) hSLE κ ( ν ) for κ ∈ ( 0 , 8 ) and ν ∈ R . driving function : dW t = √ κ dB t + κ∂ x 1 log Z κ,ν ( W t , V 2 t , V 3 t , V 4 t ) dt . Hao Wu (THU) Hypergeometric SLE 14 / 32

Hypergeometric SLE General Boundary Conditions Thm [Izyurov 2014, W. 2017] The interface of critical Ising model on Z 2 with alternating boundary condition converges to Hypergeometric SLE 3 , denoted by hSLE 3 . Q1 : What is Hypergeometric SLE? Q2 : Why they are the limit? Q3 : How to prove the convergence? Answer to Q1 : when ν = − 2, it equals SLE κ when κ ∈ ( 4 , 8 ) , SLE κ in Ω from x 1 to x 4 conditioned to avoid ( x 2 , x 3 ) is hSLE κ ( κ − 6 ) reversibility : the time-reversal has the same law. � proved for ν ≥ κ/ 2 − 4;? should be true for ν > − 4 ∨ ( κ/ 2 − 6 ) . Hao Wu (THU) Hypergeometric SLE 15 / 32

Hypergeometric SLE Q2 : Why they are the limit? Recall : Conformal Invariance + Domain Markov Property → SLE ( κ ) . Assume the scaling limit exists, then the limit should satisfy (CI) Conformal Invariance (DMP) Domain Markov Property (SYM) Symmetry Thm [W.2017] Suppose ( P q , q ∈ Q ) is a collection of proba, measures on pairs of simple curves that satisfies CI, DMP , and SYM. Then there exist κ ∈ ( 0 , 4 ] and ν < κ − 6 such that P q ∼ hSLE κ ( ν ) . Key in the proof : J. Dubédat’s commutation relation. Hao Wu (THU) Hypergeometric SLE 16 / 32

Hypergeometric SLE Q3 : How to prove the convergence? y L y R ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ Proposition ⊕ ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊖ ⊕ Fix κ ∈ ( 0 , 4 ] . There exists a ⊕ ⊖ ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ unique probability measure on ( η L ; η R ) such that ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ L ( η L | η R ) = SLE κ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ ⊕ x L ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ x R ⊖ ⊖ L ( η R | η L ) = SLE κ The marginal of η R is hSLE κ from ( η L ; η R ) : any subseq. limit x R to y R . L ( η L | η R ) = SLE 3 L ( η R | η L ) = SLE 3 Conclusion η R : hSLE 3 from x R to y R . Hao Wu (THU) Hypergeometric SLE 17 / 32

Hypergeometric SLE ⊖ y R y L ⊖ y R y L η T ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊕ η L ⊕ η R ⊕ ⊖ ⊕ ⊕ η B ⊖ x L ⊖ x R x L x R hSLE 3 hSLE 3 ( − 5 ) y R y R y L free y L free η T ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ η R ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ η L ⊕ ⊖ ⊕ ⊕ ⊕ η B ⊖ ⊖ x L x R x L x R hSLE 3 ( − 3 / 2 ) hSLE 3 ( − 7 / 2 ) Hao Wu (THU) Hypergeometric SLE 18 / 32

Hypergeometric SLE Convergence of Ising Interface to hSLE 3 y L y R Dobrushin b.c. : Interface → SLE 3 ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ RSW = ⇒ tightness ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ Holomorphic observable ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ Alternating b.c. : Interface → hSLE 3 ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ ⊕ x L ⊖ ⊖ x R ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ First approach [Izyurov] Second approach [W.] RSW = ⇒ tightness RSW = ⇒ tightness New holomorphic observable Cvg with Dobrushin b.c. Advantage : Advantage : more general b.c. more general b.c. and other lattice models. Hao Wu (THU) Hypergeometric SLE 19 / 32

More complicated b.c. Table of contents Background 1 Ising model SLE Hypergeometric SLE 2 More complicated b.c. 3 Pure Partition Functions 4 Hao Wu (THU) Hypergeometric SLE 20 / 32

More complicated b.c. What about more complicated b.c.? x 1 x 6 ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ x 5 x 2 ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ x 4 x 3 Hao Wu (THU) Hypergeometric SLE 21 / 32