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Exponential bounds for random walks on hyperbolic spaces without moment conditions

Sébastien Gouëzel

Vol. 4 (2022), No. 4, 635–671
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 escape rate 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.

Keywords
large deviations, random walks, hyperbolic groups
Mathematical Subject Classification
Primary: 20F67, 20P05, 60B15, 60F10
Milestones
Received: 15 February 2021
Revised: 28 April 2022
Accepted: 1 June 2022
Published: 15 January 2023
Authors
Sébastien Gouëzel
IRMAR, CNRS UMR 6625
Université de Rennes 1
Rennes
France