# Concentration inequalities in spaces of random configurations with positive Ricci curvatures

## TL;DR

This paper establishes a concentration inequality akin to Azuma-Hoeffding for various models of random configurations with positive Ricci curvature, utilizing Ollivier's Ricci curvature framework on graphs.

## Contribution

It provides a unified, cleaner concentration inequality for Lipschitz functions on graphs with positive Ricci curvature, extending to multiple random configuration models.

## Key findings

- Proves a concentration inequality for graphs with Ricci curvature ≥ κ>0.
- Applies the inequality to Erdős-Rényi graphs, regular directed graphs, and permutations.
- Demonstrates exponential decay of deviation probabilities for Lipschitz functions.

## 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 \[\nu \left( |f-E_{\nu}f| \geq t \right) \leq 2\exp\left( -\frac{t^2\kappa}{7} \right).\]