A survey of Ricci curvature for metric spaces and Markov chains

Abstract

This text is a presentation of the general context and results of [Oll07] and [Oll09], with comments on related work. The goal is to present a notion of Ricci curvature valid on arbitrary metric spaces, such as graphs, and to generalize a series of classical theorems in positive Ricci curvature, such as spectral gap estimates, concentration of measure or log-Sobolev inequalities.

The necessary background (concentration of measure, curvature in Riemannian geometry, convergence of Markov chains) is covered in the first section. Special emphasis is put on open questions of varying difficulty.