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May 2017 Characterization of cutoff for reversible Markov chains
Riddhipratim Basu, Jonathan Hermon, Yuval Peres
Ann. Probab. 45(3): 1448-1487 (May 2017). DOI: 10.1214/16-AOP1090

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 $\infty$.

Citation

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Riddhipratim Basu. Jonathan Hermon. Yuval Peres. "Characterization of cutoff for reversible Markov chains." Ann. Probab. 45 (3) 1448 - 1487, May 2017. https://doi.org/10.1214/16-AOP1090

Information

Received: 1 December 2014; Revised: 1 November 2015; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 1374.60129
MathSciNet: MR3650406
Digital Object Identifier: 10.1214/16-AOP1090

Subjects:
Primary: 60J10

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 3 • May 2017
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