Characterization of cutoff for reversible Markov chains

TL;DR

This paper characterizes the cutoff phenomenon in reversible Markov chains by linking it to hitting time concentration of certain sets, providing conditions and bounds for when cutoff occurs.

Contribution

It establishes a necessary and sufficient condition for cutoff in reversible chains based on hitting time concentration and relates cutoff to relaxation and mixing times on trees.

Findings

Cutoff occurs iff hitting times of certain sets concentrate.

Bounds on total variation distance are derived in terms of hitting probabilities.

On trees, cutoff is characterized by the ratio of relaxation to mixing times.

Abstract

A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of "worst" (in some sense) sets of stationary measure at least $\alpha$, for some $\alpha \in (0,1)$. We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some "worst" set of stationary measure at least $\alpha$ was not hit by time $t$. As an application of our techniques we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the ratio of their relaxation-times and their (lazy) mixing-times tends to 0.