{
  "abstract": "We compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N \\in \\mathbb{N}$ and $δ\\in \\mathbb{R} \\setminus \\{-1, -2, \\ldots\\}$, let \\[ H_N^δ := \\left( \\frac{\\sin((1+δ)π)}{π(j+k+1+δ)} \\right)_{j,k=0}^{N-1} \\] be the generalized Hilbert matrix. For $δ$ not a half-integer, we prove the large-$N$ power-law asymptotics of the determinant \\[ \\log \\text{det}(I_N \\pm H_N^δ) = -\\frac{θ_δ^2 \\pm θ_δ}{2} \\log N + O(1) \\] where $θ_δ:= δ$ if $δ< -1/2$, and $θ_δ:= \\frac 1 π\\arcsin(\\sin(δπ))$ if $δ\\ge -1/2$ and $\\arcsin \\colon [-1,1] \\to [-\\fracπ{2}, \\fracπ{2}]$ denotes the principal branch of $\\arcsin$. For $δ\\geq -\\frac 12$ the asymptotics is known. The novelty of the present paper is the regime $δ< - \\frac 12$ and understanding the dichotomy in the decay exponent. This stems from the $\\pm 1$ edge eigenvalues of the limiting operator appearing for $δ< - \\frac 12$. Most notably, we obtain for $δ< -\\frac12$ \\[ \\log \\det\\bigl(I_N - (H_N^δ)^2\\bigr) = -δ^2 \\log N + O(1). \\] Such determinants arise in the study of Anderson's orthogonality catastrophe.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00255",
  "author_total": 2,
  "authors": [
    {
      "name": "Martin Gebert"
    },
    {
      "name": "Heinrich Küttler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8q3xqkr9gkt7c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00255",
  "fulltext_url": null,
  "id": "8q3xqkr9gkt7c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00255",
  "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-ph"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7206208273",
  "pdf_url": "https://arxiv.org/pdf/2609.00255",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8q3xqkr9gkt7c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:00Z",
      "native_id": "2609.00255",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00255"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Szegő-type determinant asymptotics for generalized Hilbert matrices with edge eigenvalues",
  "title_text": "Szegő-type determinant asymptotics for generalized Hilbert matrices with edge eigenvalues",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/8q3xqkr9gkt7c",
  "venue": "arXiv",
  "versions": []
}