Sublinear deviation between geodesics and sample paths

TL;DR

This paper proves that random walk paths closely follow geodesics in hyperbolic-like spaces with sublinear deviation, with applications to Teichmüller distance and various finitely generated groups.

Contribution

It establishes a sublinear tracking property for random walks on groups acting on hyperbolic-like spaces, including Teichmüller space and Cayley graphs.

Findings

Sublinear deviation between sample paths and geodesics in hyperbolic-like spaces.

Application of sublinear tracking to Teichmüller distance for mapping class groups.

Extension of results to Cayley graphs of finitely generated groups.

Abstract

We give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic-like properties. As an application, we prove sublinear tracking in Teichmueller distance for random walks on mapping class groups, and on Cayley graphs of a large class of finitely generated groups.