A survey of Ricci curvature for metric spaces and Markov chains

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. Our starting point is the following: Is there a common geometric feature between the N -dimensional sphere SN , the discrete cube {0, 1}N , and the space RN equipped with a Gaussian measure? For a start, all three spaces exhibit the concentration of measure phenomenon; moreover, it is known (Dvoretzky theorem) that random small-dimensional sections of the cube are close to a sphere, and small-dimensional projections of either the sphere or the cube give rise to nearly-Gaussian measures. So one can wonder whether there exists a common underlying geometric property. A hint is given by the Gromov–Lévy theorem [Gro86], which states that Gaussian concentration occurs not only for the N -dimensional sphere, but for all Riemannian manifolds of positive curvature in the sense that their Ricci curvature is at least that of the sphere. In Riemannian geometry, Ricci curvature is the relevant notion in a series of positivecurvature theorems (see section 1.2). One is left with the problem of finding a definition of Ricci curvature valid for spaces more general than Riemannian manifolds. Moreover, the definition should be local and not global, since the idea of curvature is to seek local properties entailing global constraints. A first step in this direction is provided by Bakry–Émery theory [BE85], which allows to define the Ricci curvature of a diffusion process on a Riemannian manifold (or equivalently, of a second-order differential operator); when the diffusion is the ordinary Brownian motion, this gives back usual Ricci curvature. When applied to the natural process on RN associated with the Gaussian measure, this yields a positive curvature for the Gaussian space. Because the Bakry–Émery definition involves differential calculus, it is not readily adaptable to discrete spaces. To deal with the next basic example, the discrete cube, one has to drop the continuity aspect and deal with more “robust” or “coarse” notions that forget the small-scale properties of the underlying space. This is similar in spirit to what has been done for a long time in the (very different) world of negative curvature, for which coarse notions such as δ-hyperbolicity and CAT(0) spaces have been developed.