Title:
Quantitative equidistribution of random walks by automorphisms on nilmanifolds

Abstract:
Consider the random walk on a nilmanifold X induced by random automorphisms on some fixed stating point x0.
By a theorem of Bourgain-Furman-Lindenstrauss-Mozes, for X a torus and x0 irrational, the random walk approaches a uniform distribution at an exponential rate.
Benoist-Quint partially extends this to a nilmanifold, without giving any rate. Bekka-Guivarc'h shows a spectral gap exists in this setting.
I will report on an upcoming work with Weikun He and Elon Lindenstrauss that generalizes these results and our previous result on Heisenberg nilmanifolds, giving an effective rate of equidistribution from random walks by automorphisms on 2-step nilmanifolds.