{
  "abstract": "We address two questions posed by Qu and Valkó in their study of the pair correlation function of the $\\mathrm{Sine}_β$ process. First, we prove that the pair correlation function admits a jointly continuous version in the inverse-temperature and spatial parameters on $(0,\\infty)\\times\\mathbb{R}$, removing the restriction $β>2$ in their joint-continuity result. The argument uses a general observation: separate weak continuity of a family of probability laws, together with stochastic monotonicity in one parameter, implies joint weak continuity. Applied to the terminal value of the Qu--Valkó diffusion, their Palm-density formula then gives the result without differentiating the Fourier expansion. Second, for $β=2n$ we give a direct proof that the Qu--Valkó matrix-recursion power series agrees with the Selberg-integral representation recorded by Forrester. A Krawtchouk transform converts the Qu--Valkó system into a $(2n+1)$-dimensional differential system with parameter $-(n+1)$, while a centered Aomoto--Selberg trace system has parameter $n+1$. An explicit triangular differential intertwiner reflects the parameter $η\\mapsto-η$. Matching the unique Frobenius branch of exponent $2$ gives exactly the Selberg normalization. As a consequence, every coefficient of the Qu--Valkó recursion is identified with an even centered moment of the corresponding Jacobi--Selberg trace statistic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17596",
  "author_total": 1,
  "authors": [
    {
      "name": "Weiyang Fang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ajwfftwed6w6w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17596",
  "fulltext_url": null,
  "id": "ajwfftwed6w6w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17596",
  "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.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7213521729",
  "pdf_url": "https://arxiv.org/pdf/2609.17596",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ajwfftwed6w6w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:01Z",
      "native_id": "2609.17596",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17596"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Joint Continuity and Selberg--ODE Equivalence for the $\\mathrm{Sine}_β$ Pair Correlation Function",
  "title_text": "Joint Continuity and Selberg--ODE Equivalence for the Sine_β Pair Correlation Function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/ajwfftwed6w6w",
  "venue": "arXiv",
  "versions": []
}