A product chain without cutoff

TL;DR

This paper constructs a specific sequence of product chains that do not exhibit the cutoff phenomenon in total-variation or separation distances, but do show pre-cutoff, highlighting nuanced behaviors in Markov chain mixing times.

Contribution

It provides the first explicit example of product chains lacking cutoff, advancing understanding of mixing time phenomena in Markov chains.

Findings

Product chains can lack cutoff in both total-variation and separation distances.

Such chains necessarily exhibit pre-cutoff behavior.

The example clarifies the distinction between cutoff and pre-cutoff in Markov chains.

Abstract

In this note, we construct an example of a sequence of $n$-fold product chains which does not display cutoff for total-variation distance neither for separation distance. In addition we show that this type of product chains necessarily displays pre-cutoff.