Regularized modified log-Sobolev inequalities, and comparison of Markov chains

TL;DR

This paper introduces a regularized MLSI comparison method for reversible Markov chains, providing sharp estimates for the switch chain's mixing time on bipartite graphs, resolving a long-standing open problem.

Contribution

We develop a regularized MLSI comparison technique that simplifies estimating MLSI constants and apply it to determine the optimal mixing time of the switch chain on bipartite graphs.

Findings

Sharp MLSI constant estimate for the switch chain

Mixing time of order O_d(n log n) for bipartite graphs

Resolution of a long-standing open problem

Abstract

In this work, we develop a comparison procedure for the Modified log-Sobolev Inequality (MLSI) constants of two reversible Markov chains on a finite state space. Efficient comparison of the MLSI Dirichlet forms is a well known obstacle in the theory of Markov chains. We approach this problem by introducing a {\it regularized} MLSI constant which, under some assumptions, has the same order of magnitude as the usual MLSI constant yet is amenable for comparison and thus considerably simpler to estimate in certain cases. As an application of this general comparison procedure, we provide a sharp estimate of the MLSI constant of the switch chain on the the set of simple bipartite regular graphs of size $n$ with a fixed degree $d$. Our estimate implies that the total variation mixing time of the switch chain is of order $O_d(n\log n)$. The result is optimal up to a multiple depending on $d$ and resolves a long-standing open problem. We expect that the MLSI comparison technique implemented in this paper will find further applications.