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.
[1]
Peter Winkler,et al.
Mixing Points on a Circle
,
2005,
APPROX-RANDOM.
[2]
Gunnar Blom,et al.
Problems and Snapshots from the World of Probability
,
1993
.
[3]
Michael W. Mahoney,et al.
Rapid Mixing of Several Markov Chains for a Hard-Core Model
,
2003,
ISAAC.
[4]
W. Krauth.
Statistical Mechanics: Algorithms and Computations
,
2006
.
[5]
Bernd A. Berg,et al.
Markov chain monte carlo simulations and their statistical analysis: with web-based fortran code
,
2004
.
[6]
F. Martinelli.
Lectures on Glauber dynamics for discrete spin models
,
1999
.
[7]
L. Devroye.
Non-Uniform Random Variate Generation
,
1986
.
[8]
Peter Winkler,et al.
Mixing Points on an Interval
,
2005,
ALENEX/ANALCO.