Mixing and perfect sampling in one-dimensional particle systems

Laboratoire de Physique Statistique, Département de Physique de l'ENS, Ecole Normale Supérieure, PSL Research University, Université Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Univ. Paris 06, CNRS - 75005 Paris, France, and Max-Planck-Institut für Physik komplexer Systeme - Nöthnitzer Str. 38, 01187 Dresden, Germany

^(a) ze.lei@ens.fr
^(b) werner.krauth@ens.fr

Received: 18 June 2018
Accepted: 12 October 2018

Abstract

We study the approach to equilibrium of the event-chain Monte Carlo (ECMC) algorithm for the one-dimensional hard-sphere model. Using the connection to the coupon-collector problem, we prove that a specific version of this local irreversible Markov chain realizes perfect sampling in single steps, whereas the reversible local Metropolis algorithm requires single steps for mixing. This confirms a special case of an earlier conjecture about scaling of mixing times of ECMC and of the lifted forward Metropolis algorithm, its discretized variant. We also prove that sequential ECMC (with swaps) realizes perfect sampling in single events. Numerical simulations indicate a cross-over towards mixing for the sequential forward swap Metropolis algorithm, that we introduce here. We point out open mathematical questions and possible applications of our findings to higher-dimensional models.