Bounds for the random walk speed in terms of the Teichmüller distance

Abstract

Consider a closed surface $S$ with negative Euler characteristic, and an admissible probability measure on the fundamental group of $S$ with finite first moment with respect to some hyperbolic metric on $S$. Corresponding to each point in Teichmüller space there is an associated random walk on the hyperbolic plane. Azemar--Gadre--Gouëzel--Haettel--Lessa--Uyanik prove that the drift of this random walk is a proper function on Teichmüller space, and that this drift grows at least linearly with respect to the Teichmüller distance. In this paper we refine the result. On the one hand, by considering Jenkins-Strebel directions we show that the linear lower bound is sharp. On the other hand, we show that for Lebesgue typical Teichmüller geodesics, the drift grows exponentially. We also exhibit Teichmüller geodesics for which the growth oscillates between almost linear and exponential. Furthermore, we show that the drift is a quasiconvex function up to a multiplicative constant.