{"paper":{"title":"Tail decay for the distribution of the endpoint of a directed polymer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.PR","nlin.SI"],"primary_cat":"math-ph","authors_text":"Karl Liechty, Thomas Bothner","submitted_at":"2012-12-16T18:48:15Z","abstract_excerpt":"We obtain an asymptotic expansion for the tails of the random variable $\\tcal=\\arg\\max_{u\\in\\mathbb{R}}(\\mathcal{A}_2(u)-u^2)$ where $\\mathcal{A}_2$ is the Airy$_2$ process. Using the formula of Schehr \\cite{Sch} that connects the density function of $\\tcal$ to the Hastings-McLeod solution of the second Painlev\\'e equation, we prove that as $t\\rightarrow\\infty$, $\\mathbb{P}(|\\tcal|>t)=Ce^{-4/3\\varphi(t)}t^{-145/32}(1+O(t^{-3/4}))$, where $\\varphi(t)=t^3-2t^{3/2}+3t^{3/4}$, and the constant $C$ is given explicitly."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.3816","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"}