Total Variation and Separation Cutoffs are not equivalent and neither one implies the other
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 totalvariation 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 totalvariation 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 arXiv:
 arXiv:1508.03913
 Bibcode:
 2015arXiv150803913H
 Keywords:

 Mathematics  Probability
 EPrint:
 37 pages, 9 figures. Details added for some proofs and minor corrections