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

Aitor Azemar; Vaibhav Gadre; Sébastien Gouëzel; Thomas Haettel; Pablo Lessa; Caglar Uyanik

doi:10.3934/jmd.2023022

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arXiv · 2212.06581

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

Aitor Azemar·Vaibhav Gadre·Sébastien Gouëzel·Thomas Haettel·Pablo Lessa·Caglar Uyanik

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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.

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Aitor Azemar, Vaibhav Gadre, Sébastien Gouëzel, Thomas Haettel, Pablo Lessa, Caglar Uyanik. 2022-12-13. Random walk speed is a proper function on Teichm\"uller space. https://doi.org/10.3934/jmd.2023022

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