{"paper":{"title":"Exponential decay of connection probabilities for subcritical Voronoi percolation in $\\mathbb{R}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Aran Raoufi, Hugo Duminil-Copin, Vincent Tassion","submitted_at":"2017-05-22T20:07:25Z","abstract_excerpt":"We prove that for Voronoi percolation on $\\mathbb{R}^d$, there exists $p_c\\in[0,1]$ such that\n  - for $p<p_c$, there exists $c_p>0$ such that $\\mathbb{P}_p[0\\text{ connected to distance }n]\\leq \\exp(-c_p n)$,\n  - there exists $c>0$ such that for $p>p_c$, $\\mathbb{P}_p[0\\text{ connected to }\\infty]\\geq c(p-p_c)$.\n  For dimension 2, this result offers a new way of showing that $p_c(2)=1/2$. This paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.07978","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"}