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Random Cluster Dynamics for the Ising Model Is Rapidly Mixing Random Cluster Dynamics for the Ising model is Rapidly Mixing Heng Guo (Joint work with Mark Jerrum) Durham Jul 29 2017 Queen Mary, University of London 1 The Model The random cluster model (Fortuin, Kasteleyn 69) Parameters 0 6 p 6 1 (edge weight), q > 0 (cluster weight). Given graph G = (V, E), the measure on subgraph r ⊆ E is defined as jrj jEnrj κ(r) πRC(r) / p (1 - p) q , where κ(r) is the number of connected components in (V, r). (1 - p)4q4 p2(1 - p)2q2 p4q 2 The random cluster model (Fortuin, Kasteleyn 69) The partition function (normalizing factor): X jrj jEnrj κ(r) ZRC(p, q) = p (1 - p) q . r⊆E Equivalent to the Tutte polynomial ZTutte(x, y): 1 q = (x - 1)(y - 1) p = 1 - y 3 The random cluster model (Fortuin, Kasteleyn 69) jrj jEnrj κ(r) πRC(r) / p (1 - p) q The motivation is to unify: • Ising model q = 2 • Potts model q > 2, integer • Bond percolation q = 1 (On Kn, Erdős-Rényi random graph) ! ! q ! • Electrical network q 0 (Spanning trees if p 0 and p 0) 4 The random cluster model (Fortuin, Kasteleyn 69) jrj jEnrj κ(r) πRC(r) / p (1 - p) q The motivation is to unify: • Ising model q = 2 • Potts model q > 2, integer • Bond percolation q = 1 (On Kn, Erdős-Rényi random graph) ! ! q ! • Electrical network q 0 (Spanning trees if p 0 and p 0) 4 The random cluster model (Fortuin, Kasteleyn 69) jrj jEnrj κ(r) πRC(r) / p (1 - p) q The motivation is to unify: • Ising model q = 2 • Potts model q > 2, integer • Bond percolation q = 1 (On Kn, Erdős-Rényi random graph) ! ! q ! • Electrical network q 0 (Spanning trees if p 0 and p 0) 4 The random cluster model (Fortuin, Kasteleyn 69) jrj jEnrj κ(r) πRC(r) / p (1 - p) q The motivation is to unify: • Ising model q = 2 • Potts model q > 2, integer • Bond percolation q = 1 (On Kn, Erdős-Rényi random graph) ! ! q ! • Electrical network q 0 (Spanning trees if p 0 and p 0) 4 The random cluster model (Fortuin, Kasteleyn 69) jrj jEnrj κ(r) πRC(r) / p (1 - p) q The motivation is to unify: • Ising model q = 2 • Potts model q > 2, integer • Bond percolation q = 1 (On Kn, Erdős-Rényi random graph) ! ! q ! • Electrical network q 0 (Spanning trees if p 0 and p 0) 4 The random cluster model (Fortuin, Kasteleyn 69) jrj jEnrj κ(r) πRC(r) / p (1 - p) q The motivation is to unify: • Ising model q = 2 • Potts model q > 2, integer • Bond percolation q = 1 (On Kn, Erdős-Rényi random graph) ! ! q ! • Electrical network q 0 (Spanning trees if p 0 and p 0) 4 Glauber dynamics Glauber dynamics (single edge update) PRC (Metropolis): Current state x ⊆ E 1. With prob. 1=2 do nothing. (Lazy) 2. Otherwise, choose an edge e u.a.r. { } 3. Move to y = x ⊕ feg with prob. min 1, πRC(y) . πRC(x) Detailed balance: π(x)P(x, y) = π(y)P(y, x) = minfπ(x), π(y)g 5 Glauber dynamics Glauber dynamics (single edge update) PRC (Metropolis): 8 { } > 1 min 1, πRC(y) if jx ⊕ yj = 1; < 2m πRC(x) { } P ⊕f g P (x, y) = 1 min , πRC(x e ) ; RC >1 - 2m e2E 1 π (x) if x = y :> RC 0 otherwise. We are interested in: { } jj t · jj 6 Tmix(PRC) = min t : PRC(x0, )- π TV ϵ , 1 Trel(PRC) = . 1 - λ2(PRC) 6 A simple example Let p < 1=2. { } π (x [ feg) min 1, RC 8 πRC(x) < p if e is not a cut edge = 1-p : p q(1-p) if e is a cut edge 7 A simple example Let p < 1=2. { } π (x [ feg) min 1, RC 8 πRC(x) < p if e is not a cut edge = 1-p : p q(1-p) if e is a cut edge 7 A simple example Let p < 1=2. { } π (x [ feg) min 1, RC 8 πRC(x) < p if e is not a cut edge = 1-p : p q(1-p) if e is a cut edge 7 A simple example Let p < 1=2. { } π (x [ feg) min 1, RC 8 πRC(x) < p if e is not a cut edge = 1-p : p q(1-p) if e is a cut edge 7 A simple example Let p < 1=2. { } π (x [ feg) min 1, RC 8 πRC(x) < p if e is not a cut edge = 1-p : p q(1-p) if e is a cut edge 7 Previous results Previous results focus on special graphs. • On the complete graph (mean-field): [Gore, Jerrum 99] [Blanca, Sinclair 15] • On the 2D lattice Z2: [Borgs et al. 99] [Blanca, Sinclair 16] [Gheissari, Lubetzky 16] q > 2: Slow mixing for the complete graph. 0 6 q 6 2: No known fast mixing bound for general graphs. 8 Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2, 6 4 2 Trel(PRC) 8n m , 6 4 2 -1 -1 Tmix(PRC) 8n m (ln πRC(x0) + ln ϵ ). (n = #vertices, m = #edges.) • For q > 2, there exists p such that Tmix(PRC) is exponential on complete graphs. [Gore, Jerrum 99] [Blanca, Sinclair 15] [Gheis- sari, Lubetzky, Peres 17] • For q > 2 and 0 < p < 1, it is #BIS-hard to approximate ZRC(p, q). [Goldberg, Jerrum 12] • For 0 6 q < 2, there is no known obstacle. 9 Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2, 6 4 2 Trel(PRC) 8n m , 6 4 2 -1 -1 Tmix(PRC) 8n m (ln πRC(x0) + ln ϵ ). (n = #vertices, m = #edges.) • For q > 2, there exists p such that Tmix(PRC) is exponential on complete graphs. [Gore, Jerrum 99] [Blanca, Sinclair 15] [Gheis- sari, Lubetzky, Peres 17] • For q > 2 and 0 < p < 1, it is #BIS-hard to approximate ZRC(p, q). [Goldberg, Jerrum 12] • For 0 6 q < 2, there is no known obstacle. 9 Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2, 6 4 2 Trel(PRC) 8n m , 6 4 2 -1 -1 Tmix(PRC) 8n m (ln πRC(x0) + ln ϵ ). (n = #vertices, m = #edges.) • For q > 2, there exists p such that Tmix(PRC) is exponential on complete graphs. [Gore, Jerrum 99] [Blanca, Sinclair 15] [Gheis- sari, Lubetzky, Peres 17] • For q > 2 and 0 < p < 1, it is #BIS-hard to approximate ZRC(p, q). [Goldberg, Jerrum 12] • For 0 6 q < 2, there is no known obstacle. 9 Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2, 6 4 2 Trel(PRC) 8n m , 6 4 2 -1 -1 Tmix(PRC) 8n m (ln πRC(x0) + ln ϵ ). (n = #vertices, m = #edges.) • For q > 2, there exists p such that Tmix(PRC) is exponential on complete graphs. [Gore, Jerrum 99] [Blanca, Sinclair 15] [Gheis- sari, Lubetzky, Peres 17] • For q > 2 and 0 < p < 1, it is #BIS-hard to approximate ZRC(p, q). [Goldberg, Jerrum 12] • For 0 6 q < 2, there is no known obstacle. 9 Swandsen-Wang algorithm Ferromagnetic Ising model (Ising, Lenz 25) A configuration σ : V ! f , g. Parameter β > 1. j j w(σ) = β mono(σ) Gibbs distribution: π(σ) ∼ w(σ). P Partition function: ZIsing(β) = σ w(σ). β4 β2 β0 2 C Exact evaluation of ZIsing is #P-hard even for β unless β = 0, 1, i. FPRAS for ZIsing for β > 1 [Jerrum, Sinclair 93] Efficient sampling [Randall, Wilson 99] 10 Ferromagnetic Ising model (Ising, Lenz 25) A configuration σ : V ! f , g. Parameter β > 1. j j w(σ) = β mono(σ) Gibbs distribution: π(σ) ∼ w(σ). P Partition function: ZIsing(β) = σ w(σ). β4 β2 β0 2 C Exact evaluation of ZIsing is #P-hard even for β unless β = 0, 1, i. FPRAS for ZIsing for β > 1 [Jerrum, Sinclair 93] Efficient sampling [Randall, Wilson 99] 10 Ferromagnetic Ising model (Ising, Lenz 25) A configuration σ : V ! f , g. Parameter β > 1. j j w(σ) = β mono(σ) Gibbs distribution: π(σ) ∼ w(σ). P Partition function: ZIsing(β) = σ w(σ). β4 β2 β0 2 C Exact evaluation of ZIsing is #P-hard even for β unless β = 0, 1, i. FPRAS for ZIsing for β > 1 [Jerrum, Sinclair 93] Efficient sampling [Randall, Wilson 99] 10 Equivalence at q = 2 1 Let β = 1-p . jEj ZIsing(β) = β ZRC (p, 2) Joint distribution on vertices and edges [Edwards, Sokal 88]: vertex colors assigned uniformly, edges chosen with prob. p, conditioned on no chosen edge is bichromatic. Marginal on vertices ) Ising model. Marginal on edges ) random cluster model. 11 Equivalence at q = 2 1 Let β = 1-p . jEj ZIsing(β) = β ZRC (p, 2) Joint distribution on vertices and edges [Edwards, Sokal 88]: vertex colors assigned uniformly, edges chosen with prob. p, conditioned on no chosen edge is bichromatic. Marginal on vertices ) Ising model. Marginal on edges ) random cluster model. 11 Equivalence at q = 2 1 Let β = 1-p . jEj ZIsing(β) = β ZRC (p, 2) Joint distribution on vertices and edges [Edwards, Sokal 88]: vertex colors assigned uniformly, edges chosen with prob. p, conditioned on no chosen edge is bichromatic. Marginal on vertices ) Ising model. Marginal on edges ) random cluster model.
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