Random walks on mapping class groups

TL;DR

This survey explores how random walks on mapping class groups relate to their actions on Teichmüller spaces and curve complexes, highlighting classical probabilistic theorems and geometric analogies.

Contribution

It provides a comprehensive overview of the probabilistic properties of random walks on mapping class groups and their geometric implications, under optimal conditions.

Findings

Analogues of laws of large numbers and central limit theorems established.

Properties of harmonic measures analyzed under optimal moment conditions.

Geometric analogy between Gromov hyperbolic spaces and Teichmüller spaces elucidated.

Abstract

This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichm\"uller spaces or curve complexes reveal the nature of random walks, and vice versa. Our emphasis is on the analogues of classical theorems, including laws of large numbers and central limit theorems, and the properties of harmonic measures, under optimal moment conditions. We also explain the geometric analogy between Gromov hyperbolic spaces and Teichm\"uller spaces that has been used to copy the properties of random walks from one to the other.