Theory of random walks on Teichmüller space

Abstract

We study random walks on the Teichmüller space of a hyperbolic surface of finite type. In particular, we establish limit laws on random walks including the laws of large numbers, central limit theorem and geodesic tracking under the optimal moment conditions. As an application, we investigate the property of a generic mapping class. In particular, we show that pseudo-Anosov mappings are exponentially generic in the mapping class group with respect to certain generating sets. Finally, we deduce analogous limit laws for random walks on other spaces that share a similar geometric property with Teichmüller space.