  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.

Statistical Mechanics