{"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"}