{
  "abstract": "Let $H_d(n,M)$ be a $d$-uniform hypergraph with $n$ vertices and $M$ hyperedges. In the critical window $M=\\frac{n}{d(d-1)}(1+λn^{-1/3})+O(1)$, its largest component $L_n$ satisfies $$n^{-2/3}L_n\\xrightarrow{\\mathcal L} \\mathcal{F}_{\\mathrm{Pit}}\\bigl((d-1)^{1/3}a,λ(d-1)^{-2/3}\\bigr).$$ (Here $\\mathcal{F}_{\\mathrm{Pit}}(x,μ)$ denotes the limiting law of the largest component of the critical random graph established by Pittel (2001).) The proof rests on two pillars. First, a bivariate exponential formula gives the exact finite-size law of $L_n$. Subsequently, cancellation of the orders $n,n^{2/3},n^{1/3}$ in the tree normalization determines the series of rooted hypertrees, then that of unicyclic components and its singular expansions at the radius of convergence $ρ_d$. These calculations are exact; they serve the fixed-excess identities and the cancellation mechanism, and nothing more: no unproved uniformity in the excess enters the proof of the limit law. Next, by exploration and martingale approximation, the ordered sizes converge unconditionally to the excursion lengths of a Brownian motion with parabolic drift. A monotone coupling transfers the law to $H_d(n,M)$. The limit law is then identified with Pittel's law by a Brownian rescaling: if $h=d-1$, it suffices to multiply lengths by $h^{-1/3}$ and evaluate at parameter $λh^{-2/3}$. The exact Poisson representation dispenses with any separate deconditioning. We also formulate a two-regime $d$-uniform Bollobas bound (Wright for fixed $\\ell$, global combinatorial beyond that), and explicitly indicate why neither regime alone suffices to close the analytic route.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27276",
  "author_total": 1,
  "authors": [
    {
      "name": "Youssouf Wirdane"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4g5xrnmbrd1xs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27276",
  "fulltext_url": null,
  "id": "4g5xrnmbrd1xs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27276",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223105472",
  "pdf_url": "https://arxiv.org/pdf/2610.27276",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4g5xrnmbrd1xs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27276",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27276"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Pittel-type limit law for the largest component of random $d$-uniform hypergraphs: enumeration and excursions",
  "title_text": "A Pittel-type limit law for the largest component of random d-uniform hypergraphs: enumeration and excursions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/4g5xrnmbrd1xs",
  "venue": "arXiv",
  "versions": []
}