Contractive coupling rates and curvature lower bounds for Markov chains
Francesco Pedrotti
TL;DR
The paper develops a unified framework linking contractive coupling rates to curvature lower bounds for finite-state Markov chains. By formulating a main inequality $\mathscr{B}(\rho,\psi) \geq K\mathscr{A}(\rho,\psi)$ with a weight function $\theta$, the authors derive entropic curvature bounds, geodesic convexity of $\phi$-entropies, and discrete Bakry–Émery curvature via the mapping representation. They apply these tools to Glauber dynamics (Ising/Curie–Weiss), Bernoulli–Laplace, hardcore models, and interacting random walks, obtaining explicit constants $K$ in terms of meeting rates and contraction parameters, and proving exponential $W_p$-contraction of the heat flow in several settings. The work bridges probabilistic coupling techniques with analytic curvature notions (entropic and coarse Ricci), extends Conforti’s approach, and supplies new curvature bounds for classical models, with localization techniques enabling treatment of infinite-state systems. It also discusses the relationships among different curvature notions (discrete coarse Ricci, entropic, Bakry–Émery) and outlines open questions connecting positive curvature to functional inequalities like MLSI.
Abstract
Contractive coupling rates have been recently introduced by Conforti as a tool to establish convex Sobolev inequalities (including modified log-Sobolev and Poincaré inequality) for some classes of Markov chains. In this work, we show how contractive coupling rates can also be used to prove stronger inequalities, in the form of curvature lower bounds for Markov chains and geodesic convexity of entropic functionals. We illustrate this in several examples discussed by Conforti, where in particular, after appropriately choosing a parameter function, we establish positive curvature in the entropic and (discrete) Bakry--Émery sense. In addition, we recall and give straightforward generalizations of some notions of coarse Ricci curvature, and we discuss some of their properties and relations with the concepts of couplings and coupling rates: as an application, we show exponential contraction of the $p$-Wasserstein distance for the heat flow in the aforementioned examples.