Transport-entropy inequalities and curvature in discrete-space Markov chains

TL;DR

This paper establishes that positive coarse Ricci curvature in discrete Markov chains implies a W^1 transport-entropy inequality for the stationary measure, and discusses potential approaches to proving a related modified log-Sobolev inequality.

Contribution

It proves a new link between coarse Ricci curvature and transport-entropy inequalities in discrete Markov chains, and explores strategies for proving a stronger MLSI conjecture.

Findings

Positive coarse Ricci curvature implies W^1 transport-entropy inequality.

Entropy interpolation approach offers a pathway to MLSI proof.

Discussion of conjectured MLSI in discrete Markov chains.

Abstract

We show that if the random walk on a graph has positive coarse Ricci curvature in the sense of Ollivier, then the stationary measure satisfies a W^1 transport-entropy inequality. Peres and Tetali have conjectured a stronger consequence, that a modified log-Sobolev inequality (MLSI) should hold, in analogy with the setting of Markov diffusions. We discuss how our entropy interpolation approach suggests a natural attack on the MLSI conjecture.