{
  "abstract": "Let $\\mathbf{A} \\sim \\mathbb{C}\\mathcal{W}_p(I,m)$ and $\\mathbf{B} \\sim \\mathbb{C}\\mathcal{W}_p(I,n)$ be two independent complex Wishart random matrices. In this work, we investigate the asymptotic distribution of the largest eigenvalue $θ_{\\max}$ of the random matrix $\\mathbf{J} = (\\mathbf{A}+\\mathbf{B})^{-1}\\mathbf{B}$. The high-dimensional regime is potentially unbalanced in the sense that, as $p \\to \\infty$, $m$ may grow faster than $p$, whereas $n$ grows proportionally to $p$. The asymptotic behavior of $θ_{\\max}$ is characterized by means of a Riemann-Hilbert (RH) problem for Jacobi polynomials with modified Jacobi weights. We establish the type 2 Tracy--Widom asymptotic distribution of $θ_{\\max}$. This provides an alternative, RH-based proof of the corresponding result of Johnstone [Ann. Statist. 36 (2008), 2638-2716] in the balanced regime, and establishes a conjecture of Jiang [Probab. Theory Relat. Fields 144 (2009), 221-246] in the unbalanced regime. In addition, we obtain an explicit correction of order $p^{-2/3}$ to the Tracy--Widom approximation, with a remainder of order $\\mathcal{O}(p^{-1})$, and show exponential decay, as a function of $p$, of the upper tail of the distribution of $θ_{\\max}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00577",
  "author_total": 4,
  "authors": [
    {
      "name": "Gustavo Didier"
    },
    {
      "name": "Aikaterini Gkogkou"
    },
    {
      "name": "John Lopez Santander"
    },
    {
      "name": "Kenneth D. T-R McLaughlin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/814rrjyxdqpdx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00577",
  "fulltext_url": null,
  "id": "814rrjyxdqpdx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00577",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00577",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/814rrjyxdqpdx.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00577",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00577"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On the asymptotic distribution of the largest eigenvalue of unbalanced complex Jacobi ensembles",
  "title_text": "On the asymptotic distribution of the largest eigenvalue of unbalanced complex Jacobi ensembles",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/814rrjyxdqpdx",
  "venue": "arXiv",
  "versions": []
}