Entropy-driven cutoff phenomena

TL;DR

This paper introduces a general method based on entropy to detect the cutoff phenomenon in Markov chains, extending understanding from birth-death chains to more complex systems, including non-reversible ones.

Contribution

It generalizes the entropy-driven approach to identify cutoff in diverse Markov chain models, including those with complex state spaces and non-reversible dynamics.

Findings

Established cutoff in various models, including non-reversible chains.

Identified drift towards stationary measure quantiles as a key factor.

Extended cutoff detection methods beyond birth-death chains.

Abstract

In this paper we present, in the context of Diaconis' paradigm, a general method to detect the cutoff phenomenon. We use this method to prove cutoff in a variety of models, some already known and others not yet appeared in literature, including a chain which is non-reversible w.r.t. its stationary measure. All the given examples clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon. In the case of birth- and-death chains this mechanism is fairly well understood; our work is an effort to generalize this picture to more general systems, such as systems having stationary measure spread over the whole state space or systems in which the study of the cutoff may not be reduced to a one-dimensional problem. In those situations the drift may be looked for by means of a suitable partitioning of the state space into classes; using a statistical mechanics language it is then possible to set up a kind of energy-entropy competition between the weight and the size of the classes. Under the lens of this partitioning one can focus the mentioned drift and prove cutoff with relative ease.