We study random walks on GL d (ℝ) whose proximal dimension r is larger than 1 and whose limit set in the Grassmannian Gr r,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.

中文翻译：

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满足舒伯特条件的线性群的随机游走

我们研究了 GL d (ℝ) 上的随机游走，其近端维度 r 大于 1 并且其在 Grassmannian Gr r,d (ℝ) 中设置的极限不包含任何舒伯特变体。这些随机游走虽然不是近端，但在许多方面表现得与近端相似。在其他结果中，我们为与这些随机游走相关的 Grassmannian 的平稳测量建立了 Hölder 型规律。使用这个和 Bourgain 离散投影定理的推广，我们证明 Bourgain-Furman-Lindenstrauss-Mozes 定理中的近端假设可以放宽到这个 Schubert 条件。