Electronic Journal of Probability

Total variation and separation cutoffs are not equivalent and neither one implies the other

Jonathan Hermon, Hubert Lacoin, and Yuval Peres

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The cutoff phenomenon describes the case when an abrupt transition occurs in the convergence of a Markov chain to its equilibrium measure. There are various metrics which can be used to measure the distance to equilibrium, each of which corresponding to a different notion of cutoff. The most commonly used are the total-variation and the separation distances. In this note we prove that the cutoff for these two distances are not equivalent by constructing several counterexamples which display cutoff in total-variation but not in separation and with the opposite behavior, including lazy simple random walk on a sequence of uniformly bounded degree expander graphs. These examples give a negative answer to a question of Ding, Lubetzky and Peres

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 44, 36 pp.

Received: 18 December 2015
Accepted: 23 May 2016
First available in Project Euclid: 26 July 2016

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Zentralblatt MATH identifier

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

Markov chains mixing time cutoff counter example

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Hermon, Jonathan; Lacoin, Hubert; Peres, Yuval. Total variation and separation cutoffs are not equivalent and neither one implies the other. Electron. J. Probab. 21 (2016), paper no. 44, 36 pp. doi:10.1214/16-EJP4687. https://projecteuclid.org/euclid.ejp/1469557137

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