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 $\mathcal{S}_{n}$.
Citation
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
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