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Suppose there exist two constants $0<h_1<h_2<1$ such that $h_1<\\frac pn<h_2.$ Then,\n  $$\\sup_{x\\in \\mathbb{R}}|\\mathbb{P}(X_n\\le x)-e^{-e^{-x}}|=\\frac{(\\log \\log n)^{2}}{2e\\log n}(1+o(1))$$ and further\n  $$ W_{1}\\left(\\mathcal{L}(X_n),\\Lambda\\right)=\\frac{(\\log\\log n)^2}{2\\log n}(1+o(1))$$\n  for $n$ large enough. Here, $\\Lambda$ is the Gumbel distribution and $\\mathcal{L}(X_n)$ is the distribution of $X_n$ with $X_n$ being some rescaled version of $\\max"},"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":"2506.16967","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-06-20T12:58:18Z","cross_cats_sorted":[],"title_canon_sha256":"4e217a7473cd5fa8aec9de9fef10593d006b32010c5ed25d770a2da8e3e66ac6","abstract_canon_sha256":"cea0c2dbf036622b23beee97178cba4a24507705a97a8d61f5b6999ed2f5c38d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:24:46.215716Z","signature_b64":"rMmUa1o8eXpcTC6tOzKabGRRFCcAcfQkwop6rruY0C3OIElTb6JoI+PZ1q1zag7Wpws853+MmCjX3JPL0QRyDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1948996874763a1c4964a1e7e869d7977aaefc725cd971a543a1f5ef4164d323","last_reissued_at":"2026-07-05T11:24:46.215232Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:24:46.215232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"How fast does spectral radius of truncated circular unitary ensemble converge?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Xujia Meng, Yutao ma","submitted_at":"2025-06-20T12:58:18Z","abstract_excerpt":"Let $z_1, \\cdots, z_p$ be the eigenvalues of $A,$ which is the left-top $p\\times p$ submatrix of an $n\\times n$ Haar-invariant unitary matrix. Suppose there exist two constants $0<h_1<h_2<1$ such that $h_1<\\frac pn<h_2.$ Then,\n  $$\\sup_{x\\in \\mathbb{R}}|\\mathbb{P}(X_n\\le x)-e^{-e^{-x}}|=\\frac{(\\log \\log n)^{2}}{2e\\log n}(1+o(1))$$ and further\n  $$ W_{1}\\left(\\mathcal{L}(X_n),\\Lambda\\right)=\\frac{(\\log\\log n)^2}{2\\log n}(1+o(1))$$\n  for $n$ large enough. Here, $\\Lambda$ is the Gumbel distribution and $\\mathcal{L}(X_n)$ is the distribution of $X_n$ with $X_n$ being some rescaled version of $\\max"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.16967","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/2506.16967/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.16967","created_at":"2026-07-05T11:24:46.215302+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.16967v1","created_at":"2026-07-05T11:24:46.215302+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.16967","created_at":"2026-07-05T11:24:46.215302+00:00"},{"alias_kind":"pith_short_12","alias_value":"DFEJS2DUOY5B","created_at":"2026-07-05T11:24:46.215302+00:00"},{"alias_kind":"pith_short_16","alias_value":"DFEJS2DUOY5BYSLE","created_at":"2026-07-05T11:24:46.215302+00:00"},{"alias_kind":"pith_short_8","alias_value":"DFEJS2DU","created_at":"2026-07-05T11:24:46.215302+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2510.07942","citing_title":"Precise convergence rate of spectral radius of product of complex Ginibre","ref_index":29,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DFEJS2DUOY5BYSLEUHT6Q2OXS5","json":"https://pith.science/pith/DFEJS2DUOY5BYSLEUHT6Q2OXS5.json","graph_json":"https://pith.science/api/pith-number/DFEJS2DUOY5BYSLEUHT6Q2OXS5/graph.json","events_json":"https://pith.science/api/pith-number/DFEJS2DUOY5BYSLEUHT6Q2OXS5/events.json","paper":"https://pith.science/paper/DFEJS2DU"},"agent_actions":{"view_html":"https://pith.science/pith/DFEJS2DUOY5BYSLEUHT6Q2OXS5","download_json":"https://pith.science/pith/DFEJS2DUOY5BYSLEUHT6Q2OXS5.json","view_paper":"https://pith.science/paper/DFEJS2DU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.16967&json=true","fetch_graph":"https://pith.science/api/pith-number/DFEJS2DUOY5BYSLEUHT6Q2OXS5/graph.json","fetch_events":"https://pith.science/api/pith-number/DFEJS2DUOY5BYSLEUHT6Q2OXS5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DFEJS2DUOY5BYSLEUHT6Q2OXS5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DFEJS2DUOY5BYSLEUHT6Q2OXS5/action/storage_attestation","attest_author":"https://pith.science/pith/DFEJS2DUOY5BYSLEUHT6Q2OXS5/action/author_attestation","sign_citation":"https://pith.science/pith/DFEJS2DUOY5BYSLEUHT6Q2OXS5/action/citation_signature","submit_replication":"https://pith.science/pith/DFEJS2DUOY5BYSLEUHT6Q2OXS5/action/replication_record"}},"created_at":"2026-07-05T11:24:46.215302+00:00","updated_at":"2026-07-05T11:24:46.215302+00:00"}