{"paper":{"title":"The growth of the infinite long-range percolation cluster","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Pieter Trapman","submitted_at":"2009-01-06T15:35:59Z","abstract_excerpt":"We consider long-range percolation on $\\mathbb{Z}^d$, where the probability that two vertices at distance $r$ are connected by an edge is given by $p(r)=1-\\exp[-\\lambda(r)]\\in(0,1)$ and the presence or absence of different edges are independent. Here, $\\lambda(r)$ is a strictly positive, nonincreasing, regularly varying function. We investigate the asymptotic growth of the size of the $k$-ball around the origin, $|\\mathcal{B}_k|$, that is, the number of vertices that are within graph-distance $k$ of the origin, for $k\\to\\infty$, for different $\\lambda(r)$. We show that conditioned on the origi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.0661","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}