The Annals of Probability

Characterization of cutoff for reversible Markov chains

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 product of their spectral gaps and their (lazy) mixing-times tends to $\infty$.

Article information

Source
Ann. Probab., Volume 45, Number 3 (2017), 1448-1487.

Dates
Revised: November 2015
First available in Project Euclid: 15 May 2017

https://projecteuclid.org/euclid.aop/1494835222

Digital Object Identifier
doi:10.1214/16-AOP1090

Mathematical Reviews number (MathSciNet)
MR3650406

Zentralblatt MATH identifier
1374.60129

Citation

Basu, Riddhipratim; Hermon, Jonathan; Peres, Yuval. Characterization of cutoff for reversible Markov chains. Ann. Probab. 45 (2017), no. 3, 1448--1487. doi:10.1214/16-AOP1090. https://projecteuclid.org/euclid.aop/1494835222

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