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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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