{"paper":{"title":"Explosion and distances in scale-free percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Julia Komjathy, Remco van der Hofstad","submitted_at":"2017-06-08T14:11:03Z","abstract_excerpt":"We investigate the weighted scale-free percolation (SFPW) model on $\\mathbb Z^d$. In the SFPW model, the vertices of $\\mathbb Z^d$ are assigned i.i.d. weights $(W_x)_{x\\in \\mathbb Z^d}$, following a power-law distribution with tail exponent $\\tau>1$. Conditioned on the collection of weights, the edges $(x,y)_{x, y \\in \\mathbb Z^d}$ are present independently with probability that a Poisson random variable with parameter $\\lambda W_x W_y/(\\|x-y\\|)^\\alpha$ is at least one, for some $\\alpha, \\lambda>0$, and where $\\|\\cdot \\|$ denotes the Euclidean distance.\n  After the graph is constructed this wa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.02597","kind":"arxiv","version":2},"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"}