On random walks on the mapping class group

TL;DR

This paper introduces an electrification of the curve graph for a surface to identify the Poisson boundary of a random walk on the mapping class group, linking it to geodesic laminations with a logarithmic moment condition.

Contribution

It presents a novel electrification technique of the curve graph to characterize the Poisson boundary in terms of geodesic laminations for the mapping class group.

Findings

Identifies the Poisson boundary with a measure on geodesic laminations

Uses electrification of the curve graph for analysis

Establishes results under a logarithmic moment condition

Abstract

We define an electrification of the curve graph of a surface S of finite type and use it to identify the Poisson boundary of a random walk on the mapping class group of S with some logarithmic moment condition as a stationary measure on the space of minimal and maximal geodesic laminations on $S$, equipped with the Hausdorff topology.