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Perfect Simulation Discussion David B. Wilson ( ) Microsoft 53 rd ISI meeting, Seoul, Korea

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Perfect Simulation Discussion David B. Wilson ( ) Microsoft 53 rd ISI meeting, Seoul, Korea

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How long to run the Markov chain? Convergence diagnostics Workhorse of MCMC Never sure of equilibration Mathematical analysis Sure of equilibration Have to be smart to get good bounds Perfect simulation Sure of equilibration Computer determines on its own how long to run Relies on special structure (Sometimes Markov chain not used)

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Perfect Simulation Methods (partial list) Asmussen-Glynn-Thorisson 92 Aldous 95 Lovász-Winkler 95 Coupling from the past (CFTP) Propp-Wilson 96 (related ideas in Letac 86, Broder 89, Aldous 90, Johnson 96) Fills algorithm (FMMR) Fill 98, Fill-Machida-Murdoch-Rosenthal 00 Cycle-popping, sink-popping Wilson 96, Propp-Wilson 98, Cohn-Propp-Pemantle 01 Dominated CFTP Kendall 98, Kendall-Møller 99 Read-once CFTP Wilson 00 Clan of ancestors Fernández-Ferrari-Garcia 00 Randomness recycler (RR) Fill-Huber 00

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Statistical Mechanics vs Statistics Many variables, homogenous and simple interactions Fewer models that get studied intensively (universality) ad hoc methods Focus on special points (phase transitions) where mixing is slow More complicated interactions More different types of models General methods to mechanize study of new models (e.g. BUGS) Focus on generic points (real world data)

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Perfect Simulation Mathematics Cycle popping algorithm used by Benjamini, Lyons, Peres, & Schramm to study uniform spanning forests on Z and other graphs CFTP used by Van den Berg & Steif to show Ising model on Z² above critical point has finitary codings CFTP used by Häggström, Jonasson, & Lyons to show that the Potts model on amenable graphs at any temperature exhibits Bernoullicity d

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Coupling methods (partial list) Monotone coupling performance guarantee, efficient if the Markov chain is Antimonotone coupling Kendall 98, Häggström-Nelander 98 Coupling for Markov random fields Häggström-Nelander 99, Huber 98 Coupling for Bayesian inference Murdoch-Green 98, Green-Murdoch 99 Slice sampling (auxillary variables) Mira-Møller-Roberts 01, Casella-Mengersen-Robert-Titterington 0x Simulated tempering (enlarges state space) (in context of perfect simulation) Møller-Nicholls 0x

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Random Tiling by Lozenges Perfect matchings on hexagonal lattice Diatomic molecules on surface Product formulas, circular boundary Monotone Markov chain

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Coupling from the past (CFTP) Run Markov chain for very long (infinitely long) time Final state is random Figure out final state

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Square-Ice model (physics) Boundary between blue & white regions visit every site once Monotone Markov chain (monotonicity not always apparent)

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Autonormal model (statistics) Gaussian free field (physics) Random height at each vertex, Guassian distribution conditional on neighboring heights Agricultural experiments Monotone Markov chain No top or bottom state

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Ising model Spins on vertices Neighboring spins prefer to be aligned Models magnetism, certain forms of brass Two different monotone Markov chains (spin & FK representations)

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Random independent set (CS) Hard-core model (physics) Set of vertices on graph, no two adjacent Monotone on bipartite graphs Even & odd sites shown in different colors

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Potts model Generalizes Ising model to multiple spins Studied extensively in physics Image restoration Monotone Markov chain (FK representation)

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Uniformly Random Spanning Tree Connected acyclic subgraph Generated via cycle- popping Also CFTP algorithm No monotonicity

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Example from stochastic geometry Impenetrable spheres model Antimonotone coupling (Kendall, Häggström- Nelander) No top state

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Fortuin-Kasteleyn (FK) model (random cluster model) 13 edges 11 missing edges 5 connected components Different qs give percolation Ising ferromagnet Potts model

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Random Planar Maps Different embeddings of graph -> different maps Enumerated by Tutte Linear time random generation by Schaeffer

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FK model on random planar maps Annealed Pick planar map G and subgraph σ together Quenched First pick planar map G Then pick subgraph σ

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Experimental values of quenched exponents 1/(νd) or 1/(2-α) β/(νd) or β/(2-α) q=2q=4q=10q=2q=4q=10 conjecture none.1452 none Janke- Johnston Schaeffer-W (preliminary)

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Torpid mixing of Swendsen-Wang for large q Complete graph q 3 Gore-Jerrum Grid graph q big Borgs-Chayes-Frieze-Kim-Tetali-Vigoda-Vu 98% cancelation

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It is also noteworthy that the q=10 measurements (and also the q=4 quenched theory predictions) violate a supposedly general bound derived by Chayes et al. [23] for quenched systems, νD>2, since νD~1.72 from the q=10 measurements. from Janke-Johnston

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Quenched exponent work still preliminary Many headaches associated with extracting exponents Many realizations of disorder, many burn-ins Torpid mixing / burn-in is one headache we dont have

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Chance favors the prepared mind. -Pasteur Most Markov chains do not have nice special properties useful for perfect simulation Special Markov chains more interesting than typical Markov chains Look for monotonicity or other features that can be used for perfect simulation, sometimes one gets lucky

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