{"paper":{"title":"Scale-free percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Gerard Hooghiemstra, Maria Deijfen, Remco van der Hofstad","submitted_at":"2011-03-01T16:21:50Z","abstract_excerpt":"We formulate and study a model for inhomogeneous long-range percolation on $\\Zbold^d$. Each vertex $x\\in\\Zbold^d$ is assigned a non-negative weight $W_x$, where $(W_x)_{x\\in\\Zbold^d}$ are i.i.d.\\ random variables. Conditionally on the weights, and given two parameters $\\alpha,\\lambda>0$, the edges are independent and the probability that there is an edge between $x$ and $y$ is given by $p_{xy}=1-\\exp\\{-\\lambda W_xW_y/|x-y|^\\alpha\\}$. The parameter $\\lambda$ is the percolation parameter, while $\\alpha$ describes the long-range nature of the model. We focus on the degree distribution in the resu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0208","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":""},"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"}