Effective equidistribution of random walks on simple homogeneous spaces

TL;DR

This paper proves that random walks on certain homogeneous spaces tend to distribute evenly over time, with exponential convergence rates under specific arithmetic conditions, by developing new methods to analyze dimensional properties of probability measures.

Contribution

It introduces a novel approach combining dimensional stability and increase techniques to analyze equidistribution, overcoming geometric obstructions in homogeneous spaces.

Findings

Random walks equidistribute unless trapped in finite sets.

Exponential convergence under arithmetic assumptions.

New methods for dimensional analysis of probability measures.

Abstract

We consider a random walk on a homogeneous space $G/\Lambda$ where $G$ is a non-compact simple Lie group and $\Lambda$ is a lattice. The walk is driven by a probability measure $\mu$ on $G$ whose support generates a Zariski-dense subgroup. We show that the random walk equidistributes toward the Haar measure unless it is trapped in a finite $\mu$-invariant set. Moreover, under arithmetic assumptions on the pair $(\Lambda, \mu)$, we show the convergence occurs at an exponential rate, tempered by the obstructions that the starting point may be high in a cusp or close to a finite orbit. The main challenge is to show that the dimensional properties of a given probability distribution on $G/\Lambda$ improve under convolution by $\mu$. For this, we develop a new method, which combines a dimensional stability result and a dimensional increase alternative. This approach allows us to bypass inherent geometric obstructions. To show dimensional stability, we establish a general subcritical projection theorem under optimal non-concentration assumptions on the projector, and a corresponding submodular inequality for irreducible representations which allows its application to random walks. Both are of independent interest. The dimensional increase alternative aligns with the spirit of Bourgain's projection theorem. It is fine-tuned for random walks and has the advantage of being valid in situations lacking transversality.