{"paper":{"title":"Critical cluster volumes in hierarchical percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Tom Hutchcroft","submitted_at":"2022-11-10T16:47:01Z","abstract_excerpt":"We consider long-range Bernoulli bond percolation on the $d$-dimensional hierarchical lattice in which each pair of points $x$ and $y$ are connected by an edge with probability $1-\\exp(-\\beta\\|x-y\\|^{-d-\\alpha})$, where $0<\\alpha<d$ is fixed and $\\beta \\geq 0$ is a parameter. We study the volume of clusters in this model at its critical point $\\beta=\\beta_c$, proving precise estimates on the moments of all orders of the volume of the cluster of the origin inside a box. We apply these estimates to prove up-to-constants estimates on the tail of the volume of the cluster of the origin, denoted $K"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.05686","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/2211.05686/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"}