Central Limit Theorems for Gromov Hyperbolic Groups

TL;DR

This paper establishes central limit theorems and laws of iterated logarithm for random walks on hyperbolic groups, extending classical results and providing new interpretations of key metrics.

Contribution

It proves new probabilistic limit theorems for hyperbolic groups and offers a novel Hilbert metric perspective on the Green metric.

Findings

Proved a central limit theorem for the drift of random walks on hyperbolic groups.

Extended previous results to a broader class of hyperbolic groups.

Provided a new Hilbert metric interpretation of the Green metric.

Abstract

In this paper we study asymptotic properties of symmetric and non-degenerate random walks on transient hyperbolic groups. We prove a central limit theorem and a law of iterated logarithm for the drift of a random walk, extending previous results by S. Sawyer and T. Steger and F. Ledrappier for certain CAT minus one groups. The proofs use a result by A. Ancona on the identification of the Martin boundary of a hyperbolic group with its Gromov boundary. We also give a new interpretation, in terms of Hilbert metrics, of the Green metric, first introduced by S. Brofferio and S. Blachere.