Cutoff for permuted Markov chains

Anna Ben-Hamou, Yuval Peres

Abstract: Let $P$ be a bistochastic matrix of size $n$, and $\Pi$ be a permutation matrix of size $n$. In this paper, we are interested in the mixing time of the Markov chain whose transition matrix is given by $Q=P\Pi$. In other words, the chain alternates between random steps governed by $P$ and deterministic steps governed by $\Pi$. We show that if the permutation $\Pi$ is chosen uniformly at random, then under mild assumptions on $P$, with high probability, the chain $Q$ exhibits cutoff at time $\frac{\log n}{h}$, where $h$ is the entropic rate of $P$. Moreover, for deterministic permutations, we improve the upper bound on the mixing time obtained by Chatterjee and Diaconis (Probab. Theory Related Fields 178 (2020) 1193–1214).