Drift of random walks on abelian covers of finite volume homogeneous spaces

TL;DR

This paper investigates the behavior of random walks on abelian covers of finite volume homogeneous spaces, focusing on drift and recurrence, with special attention to hyperbolic surfaces where unusual behaviors occur.

Contribution

It provides a comprehensive analysis of drift for random walks on abelian covers, including cases with and without cusp unfolding, and characterizes recurrence in these settings.

Findings

Drift computed for all starting points under certain conditions.

Almost everywhere drift description without cusp assumptions.

Unique behaviors observed in hyperbolic surfaces of dimension 2.

Abstract

Let $G$ be a connected simple real Lie group, $\Lambda_{0}\subseteq G$ a lattice and $\Lambda \unlhd \Lambda_{0}$ a normal subgroup such that $\Lambda_{0}/\Lambda\simeq \mathbb{Z}^d$. We study the drift of a random walk on the $\mathbb{Z}^d$-cover $\Lambda\backslash G$ of the finite volume homogeneous space $\Lambda_{0}\backslash G$. This walk is defined by a Zariski-dense compactly supported probability measure $\mu$ on $G$. We first assume the covering map $\Lambda\backslash G\rightarrow \Lambda_{0}\backslash G$ does not unfold any cusp of $\Lambda_{0}\backslash G$ and compute the drift at \emph{every} starting point. Then we remove this assumption and describe the drift almost everywhere. The case of hyperbolic manifolds of dimension 2 stands out with non-converging type behaviors. The recurrence of the trajectories is also characterized in this context.