{"paper":{"title":"Limit Profile for the Bernoulli--Laplace Urn","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Dominik Schmid, Sam Olesker-Taylor","submitted_at":"2024-09-12T10:09:47Z","abstract_excerpt":"We analyse the convergence to equilibrium of the Bernoulli--Laplace urn model: initially, one urn contains $k$ red balls and a second $n-k$ blue balls; in each step, a pair of balls is chosen uniform and their locations are switched. Cutoff is known to occur at $\\tfrac12 n \\log \\min\\{k, \\sqrt n\\}$ with window order $n$ whenever $1 \\ll k \\le \\tfrac12 n$. We refine this by determining the limit profile: a function $\\Phi$ such that \\[\n  d_\\mathsf{TV}\\bigl( \\tfrac12 n \\log \\min\\{k, \\sqrt n\\} + \\theta n \\bigr) \\to\n  \\Phi(\\theta) \\quad\\text{as}\\quad\n  n \\to \\infty \\quad\\text{for all}\\quad\n  \\theta \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.07900","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.07900/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}