{"paper":{"title":"Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dirk Erhard, Juli\\'an Mart\\'inez, Julien Poisat (CEREMADE)","submitted_at":"2013-11-12T20:20:23Z","abstract_excerpt":"We consider a continuum percolation model on $\\R^d$, $d\\geq 1$.For $t,\\lambda\\in (0,\\infty)$ and $d\\in\\{1,2,3\\}$, the occupied set is given by the union of independent Brownian paths running up to time $t$ whoseinitial points form a Poisson point process with intensity $\\lambda\\textgreater{}0$.When $d\\geq 4$, the Brownian paths are replaced by Wiener sausageswith radius $r\\textgreater{}0$.We establish that, for $d=1$ and all choices of $t$, no percolation occurs,whereas for $d\\geq 2$, there is a non-trivial percolation transitionin $t$, provided $\\lambda$ and $r$ are chosen properly.The last s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2907","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"}