{"paper":{"title":"On the Tracy-Widom$_\\beta$ Distribution for $\\beta=6$","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Alexander Its, Andrey Kapaev, Francesco Mezzadri, Tamara Grava","submitted_at":"2016-07-05T18:05:24Z","abstract_excerpt":"We study the Tracy-Widom distribution function for Dyson's $\\beta$-ensemble with $\\beta = 6$. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Vir\\'ag equation. The latter is a linear PDE which describes the Tracy-Widom functions corresponding to general values of $\\beta$. Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom $\\beta=6$ function in terms of the second Painlev\\'e transcendent and the solution of an auxiliary ODE. Rumanov also shows that this formula allows him to derive formal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.01351","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"}