The Annals of Applied Probability

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

Joris Bierkens and 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.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 2 (2017), 846-882.

Dates
Received: July 2015
Revised: March 2016
First available in Project Euclid: 26 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1495764368

Digital Object Identifier
doi:10.1214/16-AAP1217

Mathematical Reviews number (MathSciNet)
MR3655855

Zentralblatt MATH identifier
1370.60039

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 65C05: Monte Carlo methods

Keywords
Weak convergence Markov chain Monte Carlo piecewise deterministic Markov process phase transition exponential ergodicity

Citation

Bierkens, Joris; Roberts, Gareth. A piecewise deterministic scaling limit of lifted Metropolis–Hastings in the Curie–Weiss model. Ann. Appl. Probab. 27 (2017), no. 2, 846--882. doi:10.1214/16-AAP1217. https://projecteuclid.org/euclid.aoap/1495764368


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