Random walk speed is a proper function on Teichm\"uller space

TL;DR

This paper proves that the speed of a random walk on the hyperbolic plane, associated with points in Teichmüller space, is a proper function and relates its growth to Teichmüller distance, using advanced pivoting techniques.

Contribution

It establishes the properness of the random walk speed function on Teichmüller space and connects it to the Teichmüller metric, adapting Gou"ezel's pivoting methods.

Findings

Random walk speed is a proper function on Teichmüller space.

Growth of the speed correlates with Teichmüller distance.

Adaptation of pivoting techniques to hyperbolic actions.

Abstract

Consider a closed surface $M$ with negative Euler characteristic, and an admissible probability measure on the fundamental group of $M$ with finite first moment. Corresponding to each point in the Teichm\"uller space of $M$, there is an associated random walk on the hyperbolic plane. We show that the speed of this random walk is a proper function on the Teichm\"uller space of $M$, and we relate the growth of the speed to the Teichm\"uller distance to a basepoint. One key argument is an adaptation of Gou\"ezel's pivoting techniques to actions of a fixed group on a sequence of hyperbolic metric spaces.