The Annals of Probability

Characterization of cutoff for reversible Markov chains

Riddhipratim Basu, Jonathan Hermon, and Yuval Peres

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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

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

Received: December 2014
Revised: November 2015
First available in Project Euclid: 15 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Cutoff mixing-time finite reversible Markov chains hitting times trees maximal inequality


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.

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