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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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

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1469557137

Digital Object Identifier
doi:10.1214/16-EJP4687

Mathematical Reviews number (MathSciNet)
MR3530321

Zentralblatt MATH identifier
1345.60077

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

Keywords
Markov chains mixing time cutoff counter example

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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References

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