{"paper":{"title":"Nonamenable Poisson zoo","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.GR","math.MP"],"primary_cat":"math.PR","authors_text":"G\\'abor Pete, S\\'andor Rokob","submitted_at":"2025-05-11T23:08:27Z","abstract_excerpt":"In the Poisson zoo on an infinite Cayley graph $G$, we take a probability measure $\\nu$ on rooted finite connected subsets, called lattice animals, and place i.i.d. Poisson($\\lambda$) copies of them at each vertex. If the expected volume of the animals w.r.t. $\\nu$ is infinite, then the whole $G$ is covered for any $\\lambda>0$. If the second moment of the volume is finite, then it is easy to see that for small enough $\\lambda$ the union of the animals has only finite clusters, while for $\\lambda$ large enough there are also infinite clusters. Here we show that:\n  1. If $G$ is a nonamenable fre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.07145","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/2505.07145/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"}