Hypergeometric SLE and Convergence of Critical Planar Ising - PowerPoint PPT Presentation

Hypergeometric SLE and Convergence of Critical Planar Ising Interfaces Hao Wu Yau Mathematical Sciences Center, Tsinghua University, China Hao Wu (THU) Hypergeometric SLE 1 / 28

Outline Ising Model and Percolation 1 SLE 2 Hypergeometric SLE 3 Hao Wu (THU) Hypergeometric SLE 2 / 28

Ising Model and Percolation Table of contents Ising Model and Percolation 1 SLE 2 Hypergeometric SLE 3 Hao Wu (THU) Hypergeometric SLE 3 / 28

Ising Model and Percolation 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 / 28

Ising Model and Percolation Ising Model ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ Ising model is the probability ⊕ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ measure of inverse temperature ⊖ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ β > 0 : ⊖ ⊖ ⊖ ⊖ a b ⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊕ µ β, G [ σ ] ∝ exp( − β H ( σ )) ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ Hao Wu (THU) Hypergeometric SLE 5 / 28

Ising Model and Percolation 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 ) . Hao Wu (THU) Hypergeometric SLE 5 / 28

Ising Model and Percolation 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 / 28

Ising Model and Percolation Percolation Site percolation on triangular lattice : each site is chosen independently to be black or white with probability p or 1 − p . When p < 1 / 2, white sites dominate. When p > 1 / 2, black sites dominate. When p = 1 / 2, critical, the system converges to something nontrivial. Hao Wu (THU) Hypergeometric SLE 6 / 28

Ising Model and Percolation Percolation Site percolation on triangular lattice : each site is chosen independently to be black or white with probability p or 1 − p . When p < 1 / 2, white sites dominate. When p > 1 / 2, black sites dominate. When p = 1 / 2, critical, the system converges to something nontrivial. Interface Conformal invariance + Domain Markov Property Hao Wu (THU) Hypergeometric SLE 6 / 28

SLE Table of contents Ising Model and Percolation 1 SLE 2 Hypergeometric SLE 3 Hao Wu (THU) Hypergeometric SLE 7 / 28

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 8 / 28

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, Kempainen, Smirnov) κ = 6 : Percolation Courtesy to Tom Kennedy. (Camia, Newman, Smirnov) Hao Wu (THU) Hypergeometric SLE 9 / 28

SLE Percolation and Critical Ising Thm [Smirnov 2000] The interface of critical site percolation on triangular lattice converges to SLE ( 6 ) . Hao Wu (THU) Hypergeometric SLE 10 / 28

SLE Percolation and Critical Ising Thm [Smirnov 2000] The interface of critical site percolation on triangular lattice converges to SLE ( 6 ) . Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov 2010] The interface of critical Ising model on Z 2 with Dobrushin boundary condition converges to SLE ( 3 ) . Hao Wu (THU) Hypergeometric SLE 10 / 28

SLE Percolation and Critical Ising Thm [Smirnov 2000] The interface of critical site percolation on triangular lattice converges to SLE ( 6 ) . Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, 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 10 / 28

SLE What does the convergence tell us about the model? Application : arm exponents. Boundary arm exponents p + n ( r, R ) = P ≈ R − α + n , R → ∞ Interior arm exponents p n ( r, R ) = P ≈ R − α n , R → ∞ Hao Wu (THU) Hypergeometric SLE 11 / 28

SLE What does the convergence tell us about the model? Application : arm exponents. Boundary arm exponents p + n ( r, R ) = P ≈ R − α + n , R → ∞ Interior arm exponents p n ( r, R ) = P ≈ R − α n , R → ∞ Q : How to calculate these exponents? Hao Wu (THU) Hypergeometric SLE 11 / 28

SLE Percolation [Lawler & Schramm & Werner, Smirnov & Werner 2000] Interior arm exponents : α n = ( n 2 − 1 ) / 12. Boundary arm exponents : α + n = n ( n + 1 ) / 6. Ising Model [W. 2016] Interior arm exponents : α 2 n = ( 16 n 2 − 1 ) / 24. Boundary arm exponents : 6 patterns b.c. ( ⊖⊕ ) b.c. ( ⊖ free ) b.c. ( freefree ) η η η ⊕ ⊕ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ free free free η η η ⊕ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ free free free α + β + γ + n ≈ n 2 / 3. n = n ( n + 1 ) / 3. n = n ( 2 n − 1 ) / 6. Hao Wu (THU) Hypergeometric SLE 12 / 28

SLE Other results on the convergence? Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov 2010] The interface of critical Ising model on Z 2 with Dobrushin boundary condition converges to SLE ( 3 ) . Hao Wu (THU) Hypergeometric SLE 13 / 28

SLE Other results on the convergence? Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov 2010] The interface of critical Ising model on Z 2 with Dobrushin boundary condition converges to SLE ( 3 ) . Different Models? Different lattices? Different Boundary Conditions? Hao Wu (THU) Hypergeometric SLE 13 / 28

SLE Other results on the convergence? Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, 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? Different Boundary Conditions? Hao Wu (THU) Hypergeometric SLE 13 / 28

SLE Other results on the convergence? Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, 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? Hao Wu (THU) Hypergeometric SLE 13 / 28

SLE Other results on the convergence? Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, 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 13 / 28

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 ) . Hao Wu (THU) Hypergeometric SLE 14 / 28

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 14 / 28