Poincar\'e, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature

TL;DR

This paper establishes bounds on spectral gap, Cheeger constant, and modified logarithmic Sobolev constant for Markov chains with non-negative entropic Ricci curvature, extending classical Riemannian geometry results to discrete spaces.

Contribution

It provides discrete analogues of classical geometric inequalities for Markov chains with non-negative Ricci curvature, depending only on the space's diameter.

Findings

Spectral gap bounded below by a constant depending on diameter

Cheeger isoperimetric constant bounded below by a diameter-dependent constant

Modified logarithmic Sobolev constant bounded below by a diameter-dependent constant

Abstract

We study functional inequalities for Markov chains on discrete spaces with entropic Ricci curvature bounded from below. Our main results are that when curvature is non-negative, but not necessarily positive, the spectral gap, the Cheeger isoperimetric constant and the modified logarithmic Sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space, with respect to a suitable metric. These estimates are discrete analogues of classical results of Riemannian geometry obtained by Li and Yau, Buser and Wang.