Aspects of Convergence of Random Walks on Finite Volume Homogeneous Spaces

TL;DR

This paper studies how random walks on finite volume homogeneous spaces converge to uniform distribution, addressing periodicity issues, convergence speed, and uniformity in ergodic cases.

Contribution

It provides new conditions for non-periodic convergence, proves exponential convergence speed from most starting points, and establishes uniform convergence properties in uniquely ergodic settings.

Findings

Periodic obstructions to convergence are characterized.

Exponential convergence speed towards Haar measure is proven.

Uniform convergence in uniquely ergodic cases is established.

Abstract

We investigate three aspects of weak* convergence of the $n$-step distributions of random walks on finite volume homogeneous spaces $G/\Gamma$ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesaro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesaro convergence towards Haar measure for uniquely ergodic random walks.