A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model

Joris Bierkens; Gareth Roberts

doi:10.1214/16-AAP1217

A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model

Joris Bierkens, Gareth Roberts

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Abstract

In Turitsyn, Chertkov and Vucelja [Phys. D 240 (2011) 410-414] a nonreversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization process in the Curie-Weiss model is derived for LMH, as well as for Metropolis-Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals n^1/2 for LMH, which should be compared to n for MH. At the critical temperature, the required jump rate equals n^3/4 for LMH and n^3/2 for MH, in agreement with experimental results of Turitsyn, Chertkov and Vucelja (2011). The scaling limit of LMH turns out to be a nonreversible piecewise deterministic exponentially ergodic "zig-zag" Markov process.

Original language	English
Pages (from-to)	846-882
Number of pages	37
Journal	Annals of Applied Probability
Volume	27
Issue number	2
DOIs	https://doi.org/10.1214/16-AAP1217
Publication status	Published - 2017

Keywords

Exponential ergodicity
Markov chain Monte Carlo
Phase transition
Piecewise deterministic Markov process
Weak convergence

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AAP1217Final published version, 374 KB

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abstract = "In Turitsyn, Chertkov and Vucelja [Phys. D 240 (2011) 410-414] a nonreversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization process in the Curie-Weiss model is derived for LMH, as well as for Metropolis-Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals n1/2 for LMH, which should be compared to n for MH. At the critical temperature, the required jump rate equals n3/4 for LMH and n3/2 for MH, in agreement with experimental results of Turitsyn, Chertkov and Vucelja (2011). The scaling limit of LMH turns out to be a nonreversible piecewise deterministic exponentially ergodic {"}zig-zag{"} Markov process.",

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A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model

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