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arXiv · 2606.05089

Quasi-isometric rigidity for random subsets in products of trees

Zhiqiang Li·Ranfeng Yu·Tianyi Zheng
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Abstract

In this article, we prove a rigidity result for quasi-isometric embeddings from a random subset $D$ of the product $\mathbb{X}$ of two regular trees into $\mathbb{X}$ itself. This can be seen as an extension of Eskin's quasi-isometric rigidity of higher-rank nonuniform lattices to random subsets. As a consequence, we give a description of the self-quasi-isometric embeddings of a random sample. We also show that two independent samples are almost surely non-quasi-isometric, confirming that such a phenomenon occurs in the higher-rank setting, as suggested by Ab\'ert. This result contrasts with the result on quasi-isometric equivalence between random sequences by Basu and Sly.

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Zhiqiang Li, Ranfeng Yu, Tianyi Zheng. 2026-06-03. Quasi-isometric rigidity for random subsets in products of trees. https://arxiv.org/abs/2606.05089

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