Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups

TL;DR

This paper establishes deviation inequalities and a Central Limit Theorem for random walks on acylindrically hyperbolic groups, providing insights into their probabilistic behavior and geometric properties.

Contribution

It introduces deviation inequalities for these groups and proves their implications, including CLTs and regularity of escape rates, extending to various classes of groups.

Findings

Deviation inequalities hold for measures with exponential tail.

Central Limit Theorem is established for random walks.

Linear bounds on variance of the walk's distance.

Abstract

We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences including Central Limit Theorems, the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point. In a second part of the paper, we show that the (exponential) deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups. These include non-elementary (relatively) hyperbolic groups, Mapping Class Groups, many groups acting on CAT(0) spaces and small cancellation groups.