In this article we define and study a stochastic process on Galoisian covers of compact manifolds. The successive positions of the process are defined recursively by picking a point uniformly in the Dirichlet domain of the previous one. We prove a theorem à la Kesten for such a process: the escape rate of the random walk is positive if and only if the cover is non amenable. We also investigate more in details the case where the deck group is Gromov hyperbolic, showing the almost sure convergence to the boundary of the trajectory as well as a central limit theorem for the escape rate.
[BMSC21] Large deviations for random walks on Gromov-hyperbolic spaces (2021) (https://arxiv.org/abs/2008.02709, to appear in Annales Scientifiques de l’École Normale Supérieure)
[CF16] Regularity Theory for Local and Nonlocal Minimal Surfaces: An Overview, Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions (Lecture Notes in Mathematics), Volume 2186, Springer; Fondazione CIME, 2016, pp. 117-158 | DOI | Zbl
[GdlH90] Hyperbolic groups, Progress in Mathematics, 83, Birkhäuser, 1990 | Zbl