Total Variation and Separation Cutoffs are not equivalent and neither one implies the other

TL;DR

This paper demonstrates that cutoff phenomena in total variation and separation distances are not equivalent by providing counterexamples, including lazy random walks on expander graphs, thus answering a previously open question.

Contribution

It proves that cutoff in total variation does not imply cutoff in separation and vice versa, through explicit counterexamples.

Findings

Counterexamples show cutoff in total variation but not in separation

Counterexamples show cutoff in separation but not in total variation

Lazy random walks on expander graphs exhibit these phenomena

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.