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2016 Total variation and separation cutoffs are not equivalent and neither one implies the other
Jonathan Hermon, Hubert Lacoin, Yuval Peres
Electron. J. Probab. 21(none): 1-36 (2016). DOI: 10.1214/16-EJP4687

Abstract

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

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Jonathan Hermon. Hubert Lacoin. Yuval Peres. "Total variation and separation cutoffs are not equivalent and neither one implies the other." Electron. J. Probab. 21 1 - 36, 2016. https://doi.org/10.1214/16-EJP4687

Information

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

zbMATH: 1345.60077
MathSciNet: MR3530321
Digital Object Identifier: 10.1214/16-EJP4687

Subjects:
Primary: 60J10

JOURNAL ARTICLE
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