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Random Cluster Dynamics for the Ising model is Rapidly Mixing Heng Guo Queen Mary, University of London Joint work with Mark Jerrum Oxford Nov 03 2016 Heng Guo (QMUL) Random Cluster 2016/11/03 1 / 41 The model and its dynamics Heng Guo

  1. Random Cluster Dynamics for the Ising model is Rapidly Mixing Heng Guo Queen Mary, University of London Joint work with Mark Jerrum Oxford Nov 03 2016 Heng Guo (QMUL) Random Cluster 2016/11/03 1 / 41

  2. The model and its dynamics Heng Guo (QMUL) Random Cluster 2016/11/03 2 / 41

  3. The random cluster model [Fortuin, Kasteleyn 1969] Parameters 0 ⩽ p ⩽ 1 (edge weight), q ⩾ 0 (cluster weight). Given graph G = ( V , E ) , the measure on subgraph r ⊆ E is defined as π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) , where κ ( r ) is the number of connected components in ( V , r ) . ( 1 − p ) 4 q 4 p 2 ( 1 − p ) 2 q 2 p 4 q Heng Guo (QMUL) Random Cluster 2016/11/03 3 / 41

  4. The random cluster model [Fortuin, Kasteleyn 1969] The partition function (normalizing factor): ∑ p | r | ( 1 − p ) | E \ r | q κ ( r ) . Z RC ( p , q ) = r ⊆ E Equivalent to the Tutte polynomial Z Tutte ( x , y ) : p = 1 − 1 q = ( x − 1 )( y − 1 ) y Heng Guo (QMUL) Random Cluster 2016/11/03 4 / 41

  5. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model Potts model Bond percolation Electrical network Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  6. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model q = 2 Potts model Bond percolation Electrical network Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  7. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation Electrical network Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  8. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation q = 1 (On K n , Erd˝ os-Rényi random graph) Electrical network Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  9. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation q = 1 (On K n , Erd˝ os-Rényi random graph) q → 0 (Spanning trees if p → 0 and q Electrical network p → 0) Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  10. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation q = 1 (On K n , Erd˝ os-Rényi random graph) q → 0 (Spanning trees if p → 0 and q Electrical network p → 0) Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  11. Glauber dynamics Glauber dynamics (single edge update) P RC (Metropolis): Current state x ⊆ E With prob. 1 / 2 do nothing. (Lazy) 1 Otherwise, choose an edge e u.a.r. 2 { } 1, π RC ( y ) Move to y = x ⊕ { e } with prob. min . 3 π RC ( x ) Detailed balance: π ( x ) P ( x , y ) = π ( y ) P ( y , x ) = min { π ( x ) , π ( y ) } Heng Guo (QMUL) Random Cluster 2016/11/03 6 / 41

  12. Glauber dynamics Glauber dynamics (single edge update) P RC (Metropolis):  { } 1 1, π RC ( y ) 2 m min if | x ⊕ y | = 1;  π RC ( x )     { } P RC ( x , y ) = 1 1, π RC ( x ⊕ { e } ) 1 − e ∈ E min if x = y ; ∑ 2 m π RC ( x )    0 otherwise.   We are interested in the mixing time τ ϵ ( P RC ) : τ ϵ ( P RC ) = min { t : || P t RC ( x 0 , · ) − π || TV ⩽ ϵ } . Heng Guo (QMUL) Random Cluster 2016/11/03 7 / 41

  13. A simple example Let p < 1 / 2. { } 1, π RC ( x ∪ { e } ) min π RC ( x )  p if e is not a cut edge   1 − p = p if e is a cut edge   q ( 1 − p ) Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

  14. A simple example Let p < 1 / 2. { } 1, π RC ( x ∪ { e } ) min π RC ( x )  p if e is not a cut edge   1 − p = p if e is a cut edge   q ( 1 − p ) Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

  15. A simple example Let p < 1 / 2. { } 1, π RC ( x ∪ { e } ) min π RC ( x )  p if e is not a cut edge   1 − p = p if e is a cut edge   q ( 1 − p ) Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

  16. A simple example Let p < 1 / 2. { } 1, π RC ( x ∪ { e } ) min π RC ( x )  p if e is not a cut edge   1 − p = p if e is a cut edge   q ( 1 − p ) Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

  17. A simple example Let p < 1 / 2. { } 1, π RC ( x ∪ { e } ) min π RC ( x )  p if e is not a cut edge   1 − p = p if e is a cut edge   q ( 1 − p ) Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

  18. Brief History Studied extensively for special graphs, such as the complete graph (mean-field) and the lattice Z 2 . Mean-field: [Gore, Jerrum 1999] [Blanca, Sinclair 2015] Z 2 : [Borgs et al. 1999] [Blanca, Sinclair 2016] [Gheissari, Lubetzky 2016] q > 2: Slow mixing for the complete graph. 0 ⩽ q ⩽ 2: No known fast mixing bound for general graphs. Heng Guo (QMUL) Random Cluster 2016/11/03 9 / 41

  19. Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2 , τ ϵ ( P RC ) ⩽ 10 n 4 m 2 ( ln π RC ( x 0 ) − 1 + ln ϵ − 1 ) . For q > 2, there exists p such that P RC is slow mixing on complete graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015] For q > 2 and 0 < p < 1, it is #BIS-hard to approximate Z RC ( p , q ) . [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle. Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

  20. Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2 , τ ϵ ( P RC ) ⩽ 10 n 4 m 2 ( ln π RC ( x 0 ) − 1 + ln ϵ − 1 ) . For q > 2, there exists p such that P RC is slow mixing on complete graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015] For q > 2 and 0 < p < 1, it is #BIS-hard to approximate Z RC ( p , q ) . [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle. Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

  21. Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2 , τ ϵ ( P RC ) ⩽ 10 n 4 m 2 ( ln π RC ( x 0 ) − 1 + ln ϵ − 1 ) . For q > 2, there exists p such that P RC is slow mixing on complete graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015] For q > 2 and 0 < p < 1, it is #BIS-hard to approximate Z RC ( p , q ) . [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle. Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

  22. Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2 , τ ϵ ( P RC ) ⩽ 10 n 4 m 2 ( ln π RC ( x 0 ) − 1 + ln ϵ − 1 ) . For q > 2, there exists p such that P RC is slow mixing on complete graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015] For q > 2 and 0 < p < 1, it is #BIS-hard to approximate Z RC ( p , q ) . [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle. Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

  23. Swandsen-Wang algorithm Heng Guo (QMUL) Random Cluster 2016/11/03 11 / 41

  24. Ferromagnetic Ising model [Ising, Lenz 1925] Parameter β > 1. A configuration σ : V → { + , − } . π Ising ( σ ) ∝ β mono ( σ ) = β m − cut ( σ ) Partition function Z Ising ( β ) = ∑ σ β mono ( σ ) β 0 β 2 β 4 Heng Guo (QMUL) Random Cluster 2016/11/03 12 / 41

  25. Equivalence at q = 2 1 Let β = 1 − p . Z Ising ( β ) = β | E | Z RC ( p , 2 ) Heng Guo (QMUL) Random Cluster 2016/11/03 13 / 41

  26. Swendsen-Wang algorithm [Swendsen, Wang 1987] A global Markov chain to sample Ising configurations. Current configuration σ Mark all monochromatic edges under σ as M 1 Remove each edge in M with probability β − 1 (Recall β − 1 = 1 − p ) 2 Assign a random spin to each component of ( V , M ) 3 Practically very fast for the Ising model, but difficult to analyze. Conjectured to be rapidly mixing for all graphs. (Open problem since 90s.) Heng Guo (QMUL) Random Cluster 2016/11/03 14 / 41

  27. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

  28. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

  29. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

  30. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

  31. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

  32. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

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