{"paper":{"title":"Universality and time-scale invariance for the shape of planar L\\'evy processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Julien Randon-Furling","submitted_at":"2014-02-05T17:54:08Z","abstract_excerpt":"For a broad class of planar Markov processes, viz. L\\'evy processes satisfying certain conditions (valid \\textit{eg} in the case of Brownian motion and L\\'evy flights), we establish an exact, universal formula describing the shape of the convex hull of sample paths. We show indeed that the average number of edges joining paths' points separated by a time-lapse $\\Delta \\tau \\in \\left[\\Delta \\tau _1, \\Delta \\tau_2\\right]$ is equal to $2\\ln \\left(\\Delta \\tau_2 / \\Delta \\tau_1 \\right)$, regardless of the specific distribution of the process's increments and regardless of its total duration $T$. Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.1105","kind":"arxiv","version":3},"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"}