Limit Profiles for Reversible Markov Chains
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 -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].
Cite
@article{arxiv.2005.13437,
title = {Limit Profiles for Reversible Markov Chains},
author = {Evita Nestoridi and Sam Olesker-Taylor},
journal= {arXiv preprint arXiv:2005.13437},
year = {2025}
}
Comments
v3. Minor mistake corrected in an error term in proof of Theorem B/3.1. Cutoff window shifted accordingly