A concentration theorem for the equilibrium measure of Markov chains with nonnegative coarse Ricci curvature

TL;DR

This paper establishes a concentration theorem for the invariant measure of Markov chains with nonnegative coarse Ricci curvature, showing exponential decay of measure outside neighborhoods of an attractive point, similar to diffusion processes.

Contribution

It introduces a concentration result linking coarse Ricci curvature to measure concentration for Markov chains, extending diffusion process behavior to discrete settings.

Findings

Invariant measure concentrates exponentially around attractive points

Measure outside balls decreases as exponential of double integral of curvature

Behavior parallels reversible diffusion processes on the real line

Abstract

A nonnegative coarse Ricci curvature for a Markov chain and the existence of an attractive point implies the concentration of the invariant probability measure around this point. The mass outside balls centered at the attractive point, as a function of the radius, decreases at least as fast as the exponential of a double integral of the coarse Ricci curvature. This is exactly the behaviour of the density of the reversible measure for diffusion processes on the real line.