{"paper":{"title":"Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Xujia Meng, Yutao ma","submitted_at":"2025-01-14T11:43:10Z","abstract_excerpt":"Consider the complex Ginibre ensemble, whose eigenvalues are $(\\lambda_i)_{1\\le i\\le n}$ and the spectral radius $R_n=\\max_{1\\le i\\le n}|\\lambda_i|.$ Set $X_n=\\sqrt{4 \\gamma_{n}}(R_{n}-\\sqrt{n}-\\frac12\\sqrt{\\gamma_{n}})$ and $F_n$ be its distribution function, where $\\gamma_{n}=\\log n-2\\log(\\sqrt{2\\pi}\\log n).$ It was proved in \\cite{Rider 2003} that $F_n$ converges weakly to the Gumbel distribution $\\Lambda.$ We prove in further in this paper that $$\\lim_{n\\to\\infty} \\frac{\\log n}{\\log\\log n}\\, W_1\\left(F_n, \\Lambda\\right)=2$$\n  and the Berry-Esseen bound\n  $$\\lim\\limits_{n\\to \\infty} \\frac{\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.08039","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.08039/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}