Concentration inequalities in spaces of random configurations with positive Ricci curvatures

Abstract

In this paper, we prove an Azuma-Hoeffding-type inequality in several classical models of random configurations, including the Erd\H{o}s-R\'enyi random graph models $G(n,p)$ and $G(n,M)$, the random $d$-out(in)-regular directed graphs, and the space of random permutations. The main idea is using Ollivier's work on the Ricci curvature of Markov chairs on metric spaces. Here we give a cleaner form of such concentration inequality in graphs. Namely, we show that for any Lipschitz function $f$ on any graph (equipped with an ergodic random walk and thus an invariant distribution $\nu$) with Ricci curvature at least $\kappa>0$, we have \[\nu \left( |f-E_{\nu}f| \geq t \right) \leq 2\exp\left( -\frac{t^2\kappa}{7} \right).\]