Effective equidistribution of random walks on simple homogeneous spaces
Timothée Bénard, Weikun He
TL;DR
The paper studies effective equidistribution of Zariski-dense random walks on simple homogeneous spaces X = G/Λ. It develops a dimensional bootstrap that converts initial positive dimension into near-full dimension through two pillars: dimensional stability under convolution and a supercritical decomposition that leverages a multislicing framework. Central innovations include a subcritical projection theorem under optimal non-concentration, a submodular inequality for irreducible representations, and a random-box/Morse-theory–inspired analysis of Ad(g) actions that yields exponential convergence rates under arithmetic hypotheses. The results extend equidistribution to all non-compact simple Lie groups, identify Diophantine starting points with exponential rates, and provide quantitative, scalable tools (multislicing, Brascamp–Lieb-type inequalities) that may apply beyond homogeneous settings. Overall, the work advances quantitative understanding of how random walks mix on high-dimensional homogeneous spaces, even in the presence of geometric obstructions like cusps or near-finite orbits.
Abstract
We consider a random walk on a homogeneous space $G/Λ$ where $G$ is a non-compact simple Lie group and $Λ$ is a lattice. The walk is driven by a probability measure $μ$ 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 $μ$-invariant set. Moreover, under arithmetic assumptions on the pair $(Λ, μ)$, 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 $G/Λ$ improve under convolution by $μ$. 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.