Exponential bounds for random walks on hyperbolic spaces without moment conditions

TL;DR

This paper proves that non-elementary random walks on hyperbolic spaces escape linearly with exponential error bounds, even without moment conditions, using an inductive decomposition approach.

Contribution

It establishes exponential escape bounds for hyperbolic space random walks without requiring moment conditions, a novel result in the field.

Findings

Random walks escape linearly to infinity.

Exponential error bounds are achieved without moment conditions.

Bounds are tight up to the rate of escape.

Abstract

We consider nonelementary random walks on general hyperbolic spaces. Without any moment condition on the walk, we show that it escapes linearly to infinity, with exponential error bounds. We even get such exponential bounds up to the rate of escape of the walk. Our proof relies on an inductive decomposition of the walk, recording times at which it could go to infinity in several independent directions, and using these times to control further backtracking.