Information-theoretic classification of the cutoff phenomenon in Markov processes

TL;DR

This paper classifies the cutoff phenomenon in Markov processes based on various information divergences, establishing equivalences within types and providing examples and new conditions for cutoff occurrence.

Contribution

It introduces a natural classification of divergences for cutoff analysis, proving equivalences and establishing new product conditions and results for non-reversible processes.

Findings

Cutoff phenomenon equivalence within divergence types

Examples showing cutoff in some divergences but not others

New product conditions for cutoff in non-reversible processes

Abstract

We investigate the cutoff phenomenon for Markov processes under information divergences such as $f$-divergences and R\'enyi divergences. We classify most common divergences into four types, namely $L^2$-type, $\mathrm{TV}$-type, separation-type and $\mathrm{KL}$ divergence, in which we prove that the cutoff phenomenon are equivalent and relate the cutoff time and window among members within each type. To justify that this classification is natural, we provide examples in which the family of Markov processes exhibit cutoff in one type but not in another. We also establish new product conditions in these settings for the processes to exhibit cutoff, along with new results in non-reversible or non-normal situations. The proofs rely on a functional analytic approach towards cutoff.