{"paper":{"title":"Painlev\\'e representation of Tracy-Widom$_\\beta$ distribution for $\\beta = 6$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.MP","math.PR","nlin.SI"],"primary_cat":"math-ph","authors_text":"Igor Rumanov","submitted_at":"2014-08-17T00:38:21Z","abstract_excerpt":"In arXiv:1306.2117, we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of $\\beta$. Using this general result, the case $\\beta=6$ is further considered here. This is the smallest even $\\beta$, when the corresponding Lax pair and its relation to Painlev\\'e II (PII) have not been known before, unlike cases $\\beta=2$ and $4$. It turns out that again everything can be expressed in terms of the Hastings-McLeod solution of PII. In particular, a second order nonlinear ODE for the logarithmic derivative of Tracy-Widom distribution for $\\beta=6$ involving the PII fu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3779","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"}