Upgrading MLSI to LSI for reversible Markov chains

TL;DR

This paper demonstrates that for reversible Markov chains, the modified log-Sobolev inequality can be upgraded to a log-Sobolev inequality with a manageable constant increase, leading to new bounds for Zero-Range processes and Lamplighter chains.

Contribution

It introduces a method to upgrade MLSI to LSI with a logarithmic cost and applies it to Zero-Range processes and Lamplighter chains, answering open questions.

Findings

First LSI estimate for Zero-Range processes on arbitrary graphs

Determined MLSI constant for Lamplighter chains on bounded-degree graphs

Provided negative answers to two open questions in the field

Abstract

For reversible Markov chains on finite state spaces, we show that the modified log-Sobolev inequality (MLSI) can be upgraded to a log-Sobolev inequality (LSI) at the surprisingly low cost of degrading the associated constant by $\log (1/p)$, where $p$ is the minimum non-zero transition probability. We illustrate this by providing the first log-Sobolev estimate for Zero-Range processes on arbitrary graphs. As another application, we determine the modified log-Sobolev constant of the Lamplighter chain on all bounded-degree graphs, and use it to provide negative answers to two open questions by Montenegro and Tetali (2006) and Hermon and Peres (2018). Our proof builds upon the `regularization trick' recently introduced by the last two authors.