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September 2011 Mixing times for random k-cycles and coalescence-fragmentation chains
Nathanaël Berestycki, Oded Schramm, Ofer Zeitouni
Ann. Probab. 39(5): 1815-1843 (September 2011). DOI: 10.1214/10-AOP634

Abstract

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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Nathanaël Berestycki. Oded Schramm. Ofer Zeitouni. "Mixing times for random k-cycles and coalescence-fragmentation chains." Ann. Probab. 39 (5) 1815 - 1843, September 2011. https://doi.org/10.1214/10-AOP634

Information

Published: September 2011
First available in Project Euclid: 18 October 2011

zbMATH: 1245.60006
MathSciNet: MR2884874
Digital Object Identifier: 10.1214/10-AOP634

Subjects:
Primary: 60B15 , 60J27

Keywords: Coalescence , cutoff phenomena , Mixing times , random cycles , Random transpositions

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 5 • September 2011
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