Mixing and perfect sampling in one-dimensional particle systems

TL;DR

This paper proves that the event-chain Monte Carlo algorithm achieves perfect sampling efficiently in one-dimensional hard-sphere models, outperforming reversible algorithms, with implications for higher-dimensional systems.

Contribution

It establishes rigorous mixing time bounds for ECMC and related algorithms, confirming conjectures and introducing new variants with improved efficiency.

Findings

ECMC achieves perfect sampling in O(N^2 log N) events.

Sequential ECMC with swaps achieves perfect sampling in O(N^2) events.

Numerical simulations suggest a crossover to O(N^2 log N) mixing times for certain algorithms.

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 O(N^2 log N) events, whereas the reversible local Metropolis algorithm requires O(N^3 log N) time steps for mixing. This confirms a special case of an earlier conjecture about O(N^2 log N) scaling of mixing times of ECMC and of the forward Metropolis algorithm, its discretized variant. We furthermore prove that sequential ECMC (with swaps) realizes perfect sampling in O(N^2) events. Numerical simulations indicate a cross-over towards O(N^2 log N) 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 statistical-physics models.