A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model. (English) Zbl 1370.60039

Bierkens, Joris; Roberts, Gareth

Ann. Appl. Probab. 27, No. 2, 846-882 (2017).

Summary: In [K. S. Turitsyn et al., Physica D 240, No. 4–5, 410–414 (2011; Zbl 1216.82022)] 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 [loc. cit.]. The scaling limit of LMH turns out to be a nonreversible piecewise deterministic exponentially ergodic “zig-zag” Markov process.

MSC:

60F05	Central limit and other weak theorems
60F17	Functional limit theorems; invariance principles
60J22	Computational methods in Markov chains
65C05	Monte Carlo methods
65C40	Numerical analysis or methods applied to Markov chains

Keywords:

Markov chain Monte Carlo method; Curie-Weiss model; scaling limit; weak convergence; piecewise deterministic Markov process; phase transition; exponential ergodicity

Citations:

Zbl 1216.82022