{
  "abstract": "Let $Z$ be a standard Hermite process of order $m\\ge2$ and Hurst index $H\\in(1/2,1)$. For $d<H-1/2$, let $Y$ be the variance-normalized version of the weighted integral $\\int_0^1 x^{-d}\\,\\mathrm dZ(x)$. Its right tail satisfies $ \\log\\Pr\\{Y>t\\}\\sim -\\frac12\\left(\\frac{t}{Λ_m}\\right)^{2/m}, t\\to\\infty, $ where the positive constant $Λ_m$ admits an infinite-dimensional variational representation. The same constant governs the sharp law of the iterated logarithm for weighted sums of nonlinear functions of long-memory Gaussian sequences. A general closed-form evaluation at finite order is not known. As $m\\to\\infty$ with $H$ and $d$ fixed, the normalized constants $\\sqrt{m!}\\,Λ_m$ decrease to an explicit positive function of $H$ and $d$, with relative error $O(m^{-1})$. This function also gives a lower bound for the normalized constant at every finite order. An exact Rényi variational representation at each order yields, in the high-rank limit, the entropy--energy variational problem of an attractive logarithmic gas. The limiting free energy is evaluated through the Selberg integral and the Barnes $G$ function. For $d\\le0$, convergence remains uniform up to the memory boundary $H=1/2$. After division by the square-root boundary factor $\\sqrt{2H-1}$, the high-rank and boundary limits commute. Their common value is explicit; for constant weights it equals $2^{3/2}π^3G(1/2)^8=2^{11/6}πe\\,\\mathsf A^{-12}$, where $\\mathsf A$ is the Glaisher--Kinkelin constant. The quadratic and cubic coefficients of the fixed-order expansion of the normalized constant as $H\\uparrow1$ converge, as $m\\to\\infty$, to those of the explicit limiting function. Global finite-order bounds and a concentrating-weight limit complement the main result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27261",
  "author_total": 1,
  "authors": [
    {
      "name": "Elina Moldavskaya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/297nkk83bxxcx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27261",
  "fulltext_url": null,
  "id": "297nkk83bxxcx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27261",
  "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": "W7223086389",
  "pdf_url": "https://arxiv.org/pdf/2610.27261",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/297nkk83bxxcx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:04Z",
      "native_id": "2610.27261",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27261"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "From Hermite Chaos to an Attractive Log-Gas: Explicit High-Rank Limits of Iterated-Logarithm and Tail Constants",
  "title_text": "From Hermite Chaos to an Attractive Log-Gas: Explicit High-Rank Limits of Iterated-Logarithm and Tail Constants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/297nkk83bxxcx",
  "venue": "arXiv",
  "versions": []
}