Cutoff for permuted Markov chains

TL;DR

This paper investigates the mixing time of a Markov chain combining random and deterministic steps, showing cutoff behavior for random permutations and improved bounds for deterministic ones.

Contribution

It establishes cutoff at a specific time for chains with random permutation steps and improves existing bounds for deterministic permutations.

Findings

Chains with random permutation steps exhibit cutoff at log(n)/h time.

High probability cutoff occurs under mild conditions on P.

Improved upper bounds for mixing time with deterministic permutations.

Abstract

Let $P$ be a bistochastic matrix of size $n$, and let $\Pi$ 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 $\frac{\log n}{\mathbf{h}}$, where $\mathbf{h}$ is the entropic rate of $P$. Moreover, for deterministic permutations, we improve the upper bound on the mixing time obtained by Chatterjee and Diaconis (2020).