Limit Profiles for Reversible Markov Chains

TL;DR

This paper extends Teyssier's method for approximating the distance to equilibrium in random walks, applying it to reversible Markov chains on homogeneous spaces, with improvements on several classical problems.

Contribution

It generalizes approximation lemmas to broader classes of Markov chains, enabling new bounds and insights for well-known stochastic processes.

Findings

Improved bounds for the $k$-cycle shuffle.

Enhanced analysis of the Ehrenfest urn with many urns.

Refined results for Gibbs sampling with specific priors and posteriors.

Abstract

In a recent breakthrough, Teyssier [Tey20] introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the $k$-cycle shuffle, improving results of Hough [Hou16] and Berestycki, Schramm and Zeitouni [BSZ11]; the Ehrenfest urn diffusion with many urns, improving results of Ceccherini-Silberstein, Scarabotti and Tolli [CST07]; a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, improving results of Diaconis, Khare and Saloff-Coste [DKS08].