Random walks on linear groups satisfying a Schubert condition

Abstract

We study random walks on GLd(ℝ) whose proximal dimension r is larger than 1 and whose limit set in the Grassmannian Grr,d(ℝ) is not contained any Schubert variety. These random walks, without being proximal, behave in many ways like proximal ones. Among other results, we establish a Hölder-type regularity for the stationary measure on the Grassmannian associated to these random walks. Using this and a generalization of Bourgain’s discretized projection theorem, we prove that the proximality assumption in the Bourgain-Furman-Lindenstrauss-Mozes theorem can be relaxed to this Schubert condition.