The varentropy criterion is sharp on expanders

TL;DR

This paper proves that the varentropy criterion precisely characterizes the cutoff phenomenon in sparse, fast-mixing Markov chains, advancing understanding of when abrupt convergence to equilibrium occurs.

Contribution

It demonstrates that the varentropy criterion is both necessary and sufficient for cutoff in a broad class of Markov chains, establishing its sharpness.

Findings

Cutoff phenomenon is equivalent to the varentropy criterion for sparse, fast-mixing chains.

Reversibility is not necessary for the cutoff characterization.

The varentropy approach provides a sharp condition for predicting cutoff.

Abstract

The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to the maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Despite the accumulation of many examples, this phenomenon is still far from being understood, and identifying the general conditions that trigger it has become one of the biggest challenges in the quantitative analysis of finite Markov chains. Very recently, the author proposed a general sufficient condition for the occurrence of a cutoff, based on a certain information-theoretical statistics called varentropy. In the present paper, we demonstrate the sharpness of this approach by showing that the cutoff phenomenon is actually equivalent to the varentropy criterion for all sparse, fast-mixing chains. Reversibility is not required.