Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion

Abstract

In this paper, we investigate the mixing time of the adjacent transposition shuffle for a deck of $N$ cards. We prove that around time $N^2\log N/(2{\pi }^2)$, the total variation distance to equilibrium of the deck distribution drops abruptly from $1$ to $0$, and that the separation distance has a similar behavior but with a transition occurring at time $(N^2\log N)/{\pi }^2$. This solves a conjecture formulated by David Wilson. We present also similar results for the exclusion process on a segment of length $N$ with $k$ particles.

Published at http://dx.doi.org/10.1214/15-AOP1004 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk
• Mixing times for random k-cycles and coalescence-fragmentation chains
• The mixing time for simple exclusion
• Cutoff phenomenon for the simple exclusion process on the complete graph

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