Ollivier curvature, Isoperimetry, concentration, and Log-Sobolev inequalitiy

TL;DR

This paper establishes new links between Ollivier curvature and functional inequalities on Markov chains, proving isoperimetric concentration, affirming open questions, and clarifying conditions for Log-Sobolev inequalities.

Contribution

It introduces a Laplacian separation principle, proves isoperimetric concentration inequalities, and clarifies the role of Ollivier sectional curvature in Log-Sobolev inequalities.

Findings

Proves isoperimetric concentration inequality for Markov chains with non-negative Ollivier curvature.

Answers open questions on exponential and Gaussian concentration.

Shows non-negative Ollivier sectional curvature is necessary for certain Log-Sobolev inequalities.

Abstract

We introduce a Laplacian separation principle for the the eikonal equation on Markov chains. As application, we prove an isoperimetric concentration inequality for Markov chains with non-negative Ollivier curvature. That is, every single point from the concentration profile yields an estimate for every point of the isoperimetric estimate. Applying to exponential and Gaussian concentration, we obtain affirmative answers to two open quesions by Erbar and Fathi. Moreover, we prove that the modified log-Sobolev constant is at least the minimal Ollivier Ricci curvature, assuming non-negative Ollivier sectional curvature, i.e., the Ollivier Ricci curvature when replacing the $\ell_1$ by the $\ell_\infty$ Wasserstein distance. This settles a recent open Problem by Pedrotti. We give a simple example showing that non-negative Ollivier sectional curvature is necessary to obtain a modified log-Sobolev inequality via positive Ollivier Ricci bound. This provides a counterexample to a conjecture by Peres and Tetali.