Mixing and perfect sampling in one-dimensional particle systems

Ze Lei (1, 2) , Werner Krauth (1, 2)

1 LPS - Laboratoire de Physique Statistique de l'ENS
2 MPI-PKS - Max-Planck-Institut für Physik komplexer Systeme

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.

Domaines: Mécanique statistique [cond-mat.stat-mech]

Ze Lei, Werner Krauth. Mixing and perfect sampling in one-dimensional particle systems. EPL - Europhysics Letters, 2018, 124 (2), pp.20003. ⟨10.1209/0295-5075/124/20003⟩. ⟨hal-01950833⟩