Tracking rates of random walks

TL;DR

This paper investigates how simple random walks in relatively hyperbolic groups and mapping class groups stay close to geodesics, revealing logarithmic bounds and refining divergence properties, with implications for geometric group theory.

Contribution

It establishes new bounds on the proximity of random walks to geodesics in these groups and refines the understanding of divergence in mapping class groups.

Findings

Random walks stay O(log(n))-close to geodesics in relatively hyperbolic groups.

Random walks stay O(√(n log n))-close to geodesics in mapping class groups.

Mapping class groups have quadratic divergence, refined in this work.

Abstract

We show that simple random walks on (non-trivial) relatively hyperbolic groups stay $O(\log(n))$-close to geodesics, where $n$ is the number of steps of the walk. Using similar techniques we show that simple random walks in mapping class groups stay $O(\sqrt{n\log(n)})$-close to geodesics and hierarchy paths. Along the way, we also prove a refinement of the result that mapping class groups have quadratic divergence. An application of our theorem for relatively hyperbolic groups is that random triangles in non-trivial relatively hyperbolic groups are $O(\log(n))$-thin, random points have $O(\log(n))$-small Gromov product and that in many cases the average Dehn function is subasymptotic to the Dehn function.