{"paper":{"title":"On the maximum of the C$\\beta$E field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CA","math.CV","math.MP"],"primary_cat":"math.PR","authors_text":"Joseph Najnudel, Reda Chhaibi, Thomas Madaule","submitted_at":"2016-07-01T13:42:16Z","abstract_excerpt":"In this paper, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according the Circular Beta Ensemble (C$\\beta$E). More precisely, if $X_n$ is this characteristic polynomial and $\\mathbb{U}$ the unit circle, we prove that: $$\\sup_{z \\in \\mathbb{U} } \\Re \\log X_n(z) =\n  \\sqrt{\\frac{2}{\\beta}}\n  \\left(\\log n - \\frac{3}{4} \\log \\log n + \\mathcal{O}(1) \\right)\\ ,$$ as well as an analogous statement for the imaginary part. The notation $\\mathcal{O}(1)$ means that the corresponding family of random variable"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00243","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"}