Linear Progress with Exponential Decay in Weakly Hyperbolic Groups

TL;DR

This paper studies the behavior of random walks in hyperbolic spaces, showing that exponential decay of progress can occur even without acylindrical actions, extending previous results.

Contribution

It extends the known exponential decay of progress in random walks to non-acylindrical actions on hyperbolic spaces.

Findings

Random walks converge to boundary with probability one.

Linear progress with exponential decay is established without acylindricity.

Results apply to spaces with step distributions having exponential tails.

Abstract

A random walk $w_n$ on a separable, geodesic hyperbolic metric space $X$ converges to the boundary $\partial X$ with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when (1) the step distribution has exponential tail and (2) the action on $X$ is acylindrical. We extend exponential decay to the non-acylindrical case.