{"paper":{"title":"Exact distributions of the number of distinct and common sites visited by N independent random walkers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Anupam Kundu, Gregory Schehr, Satya N. Majumdar","submitted_at":"2013-02-11T11:26:41Z","abstract_excerpt":"We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated to N independent random walkers. Using this mapping, we compute exactly their probability distributions P_N^d(S,t) and P_N^d(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S_N(t)/\\sqrt{t} \\propto 2 \\sqrt{\\log N} + \\widetilde{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.2452","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"}