# Random walks on linear groups satisfying a Schubert condition

## TL;DR

This paper investigates random walks on general linear groups with specific geometric constraints, establishing regularity properties and extending key theorems by relaxing proximality to a Schubert condition.

## Contribution

It introduces a new class of random walks satisfying a Schubert condition, proving regularity of stationary measures and generalizing a fundamental theorem in the field.

## Key findings

- Established Hölder regularity for stationary measures.
- Extended Bourgain-Furman-Lindenstrauss-Mozes theorem to non-proximal walks.
- Showed that Schubert conditions can replace proximality in key results.

## Abstract

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.