{"paper":{"title":"Extremal geometry of a Brownian porous medium","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Frank den Hollander, Jesse Goodman","submitted_at":"2012-11-15T15:45:44Z","abstract_excerpt":"The path W[0,t] of a Brownian motion on a d-dimensional torus T^d run for time t is a random compact subset of T^d. We study the geometric properties of the complement T^d \\ W[0,t] for t large and d >= 3. In particular, we show that the largest regions in this complement have a linear scale phi = [(d log t)/(d-2)kt]^{1/(d-2)}, where k is the capacity of the unit ball. More specifically, we identify the sets E for which T^d \\ W[0,t] contains a translate of phi E, and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of T^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3630","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"}