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Set $$X_n:=\\sqrt{\\log s_n}\\left(\\frac{2n \\max_{1\\le i\\le n}|\\zeta_i|^2-2\\sqrt{n(n+v)}}{\\sqrt{2n+v}}-a(s_{n})\\right)$$ with $s_n=n(n+v)/(2n+v)$ and $a(s_n)=\\sqrt{\\log s_n}-\\frac{\\log(\\sqrt{2\\pi}\\log s_n)}{\\sqrt{\\log s_n}}.$ It was proved in \\cite{JQ} that $X_n$ converges weakly to the Gumbel distribution $\\Lambda$. In this paper, we give in further that $$\\lim_{n\\to\\infty} \\frac{\\log"},"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":"2501.08661","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-01-15T08:56:02Z","cross_cats_sorted":[],"title_canon_sha256":"0ba1e2fb78287b6d8dcf1739ad648eb783a6b87671be3b65c6d7a4486037bc21","abstract_canon_sha256":"c87dc7d9eac2e1c8c9d9b65a50c4c5edaae1dda9dc806f1a75f77ac6fc5bb26f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:01:39.948228Z","signature_b64":"nEs+7kHsI9SxNdkfflbiMFvvWJ6t/C+QMbf/l7vJ4oYWRCOamFEUK9BnoMJF/7TwBN4R8IT6v1ApHZk/61Y5Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbdf770ca58ee3779d4ddb64f9427778ef9ad18556cde450f5861451e572d17f","last_reissued_at":"2026-07-05T10:01:39.947732Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:01:39.947732Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal $W_1$ and Berry-Esseen bound between the spectral radius of large Chiral non-Hermitian random matrices and Gumbel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Siyu Wang, Yutao ma","submitted_at":"2025-01-15T08:56:02Z","abstract_excerpt":"Consider the chiral non-Hermitian random matrix ensemble with parameters $n$ and $v$ and the non Hermiticity parameter $\\tau=0$ and let $(\\zeta_i)_{1\\le i\\le n}$ be its $n$ eigenvalues with positive $x$-coordinate. Set $$X_n:=\\sqrt{\\log s_n}\\left(\\frac{2n \\max_{1\\le i\\le n}|\\zeta_i|^2-2\\sqrt{n(n+v)}}{\\sqrt{2n+v}}-a(s_{n})\\right)$$ with $s_n=n(n+v)/(2n+v)$ and $a(s_n)=\\sqrt{\\log s_n}-\\frac{\\log(\\sqrt{2\\pi}\\log s_n)}{\\sqrt{\\log s_n}}.$ It was proved in \\cite{JQ} that $X_n$ converges weakly to the Gumbel distribution $\\Lambda$. 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