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Let $\\{\\sigma_i\\}_{1\\le i\\le n}$ be the eigenvalues of $X,$ and $Z_n$ be some rescaled version of $\\max_i \\Re \\sigma_i.$ It was proved that $Z_n$ converges weakly to the Gumbel distribution $\\Lambda_{\\beta}$ with distribution function $e^{-\\frac{\\beta}{2}e^{-x}}.$ We further prove that\n  $$\\sup_{x\\in \\mathbb{R}}|\\mathbb{P}(Z_n \\leq x)-e^{-\\frac{\\beta}{2}e^{-x}}|=\\frac{25\\log \\log n}{4e \\log n}(1+o(1))$$\n  and\n  $$ W_1\\left(\\mathcal{L}(Z_n), \\Lambda_{\\beta}\\right)=\\frac{25\\log \\log n}{4\\log n}(1+o(1))$$ for sufficiently larg"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2506.04560","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-06-05T02:28:39Z","cross_cats_sorted":[],"title_canon_sha256":"30a584e0298074fec945a878e73d7e9ffe650542683e0b3bd40eaeaa5d6aaf82","abstract_canon_sha256":"eb8dbe5bec81c599d6b5452fc1a675ada90e2a0c8fb2448bf714ef838b955a63"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:05:17.288442Z","signature_b64":"vsy0uO9ocNQ6N0TrHSP/vtuYEKObBEFpvt6ye74KLdPTx5NyADYy/ahy0KYJLgTuiopj50hhCpTrM9OVE8ShBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"900a5308dc06fe5275a38cba5fa384b4f83e627684a24ad21a72a70e8ba329c1","last_reissued_at":"2026-07-05T12:05:17.287735Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:05:17.287735Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Xinchen Hu, Yutao ma","submitted_at":"2025-06-05T02:28:39Z","abstract_excerpt":"Let $X$ be a real $(\\beta=1)$ or complex $(\\beta=2)$ Ginibre ensemble. 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