The Annals of Probability

Mixing times for random k-cycles and coalescence-fragmentation chains

Nathanaël Berestycki, Oded Schramm, and Ofer Zeitouni

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Let $\mathcal{S}_{n}$ be the permutation group on n elements, and consider a random walk on $\mathcal{S}_{n}$ whose step distribution is uniform on k-cycles. We prove a well-known conjecture that the mixing time of this process is (1/k)n log n, with threshold of width linear in n. Our proofs are elementary and purely probabilistic, and do not appeal to the representation theory of $\mathcal{S}_{n}$.

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Ann. Probab., Volume 39, Number 5 (2011), 1815-1843.

First available in Project Euclid: 18 October 2011

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Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60J27: Continuous-time Markov processes on discrete state spaces

Mixing times coalescence cutoff phenomena random cycles random transpositions


Berestycki, Nathanaël; Schramm, Oded; Zeitouni, Ofer. Mixing times for random k -cycles and coalescence-fragmentation chains. Ann. Probab. 39 (2011), no. 5, 1815--1843. doi:10.1214/10-AOP634.

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