{
  "abstract": "We study the Gibbs measure of $H(h) = \\Re\\sum_{p^{j} \\leq X}U_{p}^{j}/(j p^{j(σ+ i h)})$, a random field modeling large values of the log-modulus of the Riemann zeta function on and near the critical line, where the $U_{p}$ are independent and uniformly distributed on the complex unit circle. Writing $t = \\log \\log T$, the cutoff $X$ and real part $σ$ may vary with $T$ subject to $T^{1/t^Φ} \\leq X \\leq T^{1/t^ϕ}$ and $0 \\leq σ- 1/2 \\leq C_σt^Φ/\\log T$, for fixed $Φ\\geq ϕ> 1/2$ and $C_σ > 0$. On $I_θ = [ - (\\log T)^θ,(\\log T)^θ]$, define the Gibbs measure $G_{β,σ,T}^{\\mathrm{H}}(\\mathrm{d}h) := e^{βH(h)}\\mathrm{d}h/\\int_{I_θ}e^{βH(h')}\\mathrm{d}h'$. For every fixed $θ> - 1$ and inverse temperature $β> 0$, the limiting joint distributions of correlations (overlaps) among points sampled from this measure are determined after averaging over the Haar phases. At and below $β_{c}(θ) = 2\\sqrt{1 + (θ\\vee 0)}$, the limiting two-overlap distribution is concentrated at $q_{0} = |θ\\wedge 0|$. Above this threshold, it assigns masses $β_{c}(θ)/β$ and $1 - β_{c}(θ)/β$ to $q_{0}$ and $1$, respectively, and the limiting joint overlap distribution is described by a Poisson-Dirichlet process of parameter $β_{c}(θ)/β$. Approximate Ghirlanda-Guerra identities are shown to hold uniformly for bounded deterministic functions of the sampled positions. This paper is the first in a series devoted to extending these results to $\\log|ζ|$ near the critical line.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14901",
  "author_total": 1,
  "authors": [
    {
      "name": "Frédéric Ouimet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/18qgy4s3jvxyz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14901",
  "fulltext_url": null,
  "id": "18qgy4s3jvxyz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14901",
  "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.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221832751",
  "pdf_url": "https://arxiv.org/pdf/2610.14901",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/18qgy4s3jvxyz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.14901",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14901"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Overlap distributions under the Gibbs measure of a Haar model of $\\log|ζ|$ on and near the critical line",
  "title_text": "Overlap distributions under the Gibbs measure of a Haar model of log|ζ| on and near the critical line",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/18qgy4s3jvxyz",
  "venue": "arXiv",
  "versions": []
}