On Rapid mixing for random walks on nilmanifolds
Dmitry Dolgopyat, Spencer Durham, Minsung Kim
TL;DR
The paper addresses rapid mixing for random walks on nilmanifolds generated by finitely many translations, proving rapid mixing for almost all such walks under mild constraints on the number of generators.It introduces the algebraic notion of m-greatness, tying non-degeneracy of polynomial maps to Diophantine behavior and, via spectral-gap arguments, to decay of correlations.The authors establish broad 2-great results for many classical nilmanifold classes (quasi-abelian, triangular, step-3 or lower) and show that step-s manifolds are s-great, with general preservation under products and quotients; they also provide a counterexample illustrating the approach's limitations for 2-generators.Consequently, rapid mixing holds for a full-measure set of random walks generated by sufficiently many translations, with implications for CLTs and related statistical properties on nilmanifolds.
Abstract
We prove rapid mixing for almost all random walks generated by $m$ translations on an arbitrary nilmanifold under mild assumptions on the size of $m$. For several classical classes of nilmanifolds, we show $m=2$ suffices. This provides a partial answer to the question raised in \cite{D02} about the prevalence of rapid mixing for random walks on homogeneous spaces.