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

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