Electronic Journal of Probability
- Electron. J. Probab.
- Volume 21 (2016), paper no. 44, 36 pp.
Total variation and separation cutoffs are not equivalent and neither one implies the other
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
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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