Equidistribution of affine random walks on some nilmanifolds

TL;DR

This paper investigates how affine random walks distribute over certain nilmanifolds, establishing conditions for rapid equidistribution and extending previous results on toral random walks.

Contribution

It provides a quantitative equidistribution framework for affine random walks on nilmanifolds, linking failures to equidistribution to factors and removing previous assumptions on acting groups.

Findings

Failure of fast equidistribution linked to factor nilmanifolds

Results apply to Heisenberg nilmanifolds

Strengthens previous torus random walk results

Abstract

We study quantitative equidistribution in law of affine random walks on nilmanifolds, motivated by a result of Bourgain, Furman, Mozes and the third named author on the torus. Under certain assumptions, we show that a failure to having fast equidistribution is due to a failure on a factor nilmanifold. Combined with equidistribution results on the torus, this leads to an equidistribution statement on some nilmanifolds such as Heisenberg nilmanifolds. In an appendix we strengthen results of de Saxce and the first named author regarding random walks on the torus by eliminating an assumption on Zariski connectedness of the acting group.