Modern aspects of Markov chains: entropy, curvature and the cutoff phenomenon

TL;DR

This paper explores the cutoff phenomenon in Markov chains, examining its relation to entropy, curvature, and concentration, and discusses the challenges in identifying universal conditions for this abrupt convergence to equilibrium.

Contribution

It provides a self-contained introduction to the cutoff phenomenon and highlights recent connections with entropy, curvature, and concentration in Markov processes.

Findings

Cutoff phenomenon involves abrupt convergence to equilibrium in Markov chains.

Relations between cutoff, entropy, and curvature are recently uncovered.

Identifying universal conditions for cutoff remains a major challenge.

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 its maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Discovered four decades ago in the context of card shuffling, this surprising phenomenon has since then been observed in a variety of models, from random walks on groups or complex networks to interacting particle systems. It is now believed to be universal among fast-mixing high-dimensional processes. Yet, current proofs are heavily model-dependent, and identifying the general conditions that trigger a cutoff remains one of the biggest challenges in the quantitative analysis of finite Markov chains. The purpose of these lecture notes is to provide a self-contained introduction to this fascinating question, and to describe its recently-uncovered relations with entropy, curvature and concentration.