Ricci curvature of finite Markov chains via convexity of the entropy

TL;DR

This paper introduces a new Ricci curvature concept for Markov chains on discrete spaces based on entropy convexity, leading to discrete analogues of key geometric results and applications like the hypercube.

Contribution

It develops a novel Ricci curvature notion for discrete Markov chains using entropy convexity, extending continuous geometric results to discrete settings.

Findings

Discrete Ricci curvature bounds are preserved under tensorisation.

Established sharp Ricci curvature lower bound for the discrete hypercube.

Proved discrete analogues of Bakry--Emery and Otto--Villani results.

Abstract

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.