Random walks on linear groups satisfying a Schubert condition

We study random walks on GL(d)(Double-struck capital R) whose proximal dimensionris larger than 1 and whose limit set in the Grassmannian Gr(r,d)(Double-struck capital 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 Holder-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.