Random walks on linear groups satisfying a Schubert condition

We study random walks on $\mathrm{GL}_d(\mathbb{R})$ whose proximal dimension $r$ is larger than $1$ and whose limit set in the Grassmannian $\mathrm{Gr}_{r,d}(\mathbb{R})$ 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 H\"older-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.