{"paper":{"title":"Quasi-isometric rigidity for random subsets in products of trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.PR"],"primary_cat":"math.MG","authors_text":"Ranfeng Yu, Tianyi Zheng, Zhiqiang Li","submitted_at":"2026-06-03T16:54:08Z","abstract_excerpt":"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 t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05089","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.05089/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}