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

TL;DR

This paper proves that the mixing time for a random walk on permutation groups using uniform k-cycles is (1/k)n log n, using elementary probabilistic methods without representation theory.

Contribution

It confirms a well-known conjecture about the mixing time for k-cycle random walks with a simple probabilistic proof.

Findings

Mixing time is (1/k)n log n for k-cycle random walks.

Elementary probabilistic methods suffice for the proof.

Threshold of mixing time has linear width in n.

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 $\mathcal{S}_n$.