{"paper":{"title":"Hard edge tail asymptotics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Brian Rider, Jose A. Ramirez, Ofer Zeitouni","submitted_at":"2011-09-19T18:52:09Z","abstract_excerpt":"Let $\\Lambda$ be the limiting smallest eigenvalue in the general (\\beta, a)-Laguerre ensemble of random matrix theory. Here \\beta>0, a >-1; for \\beta=1,2,4 and integer a, this object governs the singular values of certain rank n Gaussian matrices. We prove that P(\\Lambda > \\lambda) = e^{- (\\beta/2) \\lambda + 2 \\gamma \\lambda^{1/2}} \\lambda^{- (\\gamma(\\gamma+1))/(2\\beta) + \\gamma/4} E (\\beta, a) (1+o(1)) as \\lambda goes to infinity, in which \\gamma = (\\beta/2) (a+1)-1 and E(\\beta, a) is a constant (which we do not determine). This estimate complements/extends various results previously availabl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4121","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"}