{"paper":{"title":"Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Alain Comtet, Julien Randon-Furling, Satya N. Majumdar","submitted_at":"2009-07-06T06:49:44Z","abstract_excerpt":"We compute exactly the mean perimeter and area of the convex hull of N independent planar Brownian paths each of duration T, both for open and closed paths. We show that the mean perimeter < L_N > = \\alpha_N, \\sqrt{T} and the mean area <A_N> = \\beta_N T for all T. The prefactors \\alpha_N and \\beta_N, computed exactly for all N, increase very slowly (logarithmically) with increasing N. This slow growth is a consequence of extreme value statistics and has interesting implication in ecological context in estimating the home range of a herd of animals with population size N."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.0921","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"}