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

By Nathanaël Berestycki, Oded Schramm, Ofer Zeitouni. 2010

51 citations recorded by the source corpus.

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 $\mathcal{S}_n$.

Subfields: Artificial Intelligence, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f4d481f82ee5447c799bd514dae766b9472f564