{
  "abstract": "We study the Hagedorn transition of the single-winding Dutta-Gopakumar unitary matrix model at finite rank $N$. Its Schur expansion is a sum of Plancherel probabilities restricted by the finite-rank wall on the number of rows, and the Vershik-Kerov-Logan-Shepp limit shape reaches this wall when the number of boxes equals $N^2/4$. Two scaling windows around the Hagedorn point resolve the wall at different scales. In the outer window the Baik-Deift-Johansson theorem gives a Tracy-Widom crossover, and summing it identifies Liu's critical free-energy coefficient with the first moment of the GUE Tracy-Widom distribution. In the inner window the same sum determines the canonical probability law of the Polyakov loop at the Hagedorn point: the normalized loop $\\mathrm{tr}\\,U/N$ is uniformly distributed on a disk of radius one half, so the order parameter does not self-average at large $N$. The intensity $|\\mathrm{tr}\\,U|^2/N^2$ has the same limiting law as the Schur degree $n/N^2$: at the Hagedorn point the canonical degree weights become asymptotically equal below the macroscopic finite-rank wall. Under the partial-deconfinement dictionary the uniform disk becomes a linear probability density for the deconfined color fraction, and its fully deconfined endpoint is the Young-diagram endpoint of Berenstein and Yan. The Tracy-Widom mean also fixes the first finite-$N$ correction to the Polyakov-loop Laplace transform and to all of its fixed radial moments. Bessel-Toeplitz evaluations up to $N = 100$ confirm the free-energy coefficient, the uniform law, and the predicted correction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12588",
  "author_total": 3,
  "authors": [
    {
      "name": "Robert de Mello Koch"
    },
    {
      "name": "Minkyoo Kim"
    },
    {
      "name": "Hyunwoo Oh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3erdn0eyy7rkg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12588",
  "fulltext_url": null,
  "id": "3erdn0eyy7rkg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12588",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7212630126",
  "pdf_url": "https://arxiv.org/pdf/2609.12588",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3erdn0eyy7rkg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2609.12588",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12588"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "The Hagedorn transition as a Plancherel wall: Tracy-Widom statistics, Polyakov loops and partial deconfinement",
  "title_text": "The Hagedorn transition as a Plancherel wall: Tracy-Widom statistics, Polyakov loops and partial deconfinement",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:19Z",
  "url": "https://research.draconitic.com/paper/3erdn0eyy7rkg",
  "venue": "arXiv",
  "versions": []
}