{
  "abstract": "Let $M$ be chosen uniformly from all matchings of a finite linear $k$-uniform hypergraph $H$, and let $\\overline{q}(H)$ be the average probability that a vertex is left uncovered. If $H$ has maximum degree $D$ and normalized average degree $β=k|E(H)|/(|V(H)|D)$, then, for every fixed $k\\geq 2$ and uniformly in the order, $\\overline{q}(H) \\leq 1-β+(βk+o_D(1))\\log\\log D/\\log D$. The underlying estimate is the order-uniform count $\\log Z(H)\\geq (|E(H)|/D)(\\log D-(k+o_D(1))\\log\\log D)$, and it also counts matchings of size $(1-o_D(1))|E(H)|/D$. We prove this by sampling edges, deleting those incident with unusually large sampled degrees, and controlling only the total deleted mass before applying the Molloy-Reed list edge-colouring theorem. For $d$-regular $H$, a quantitative version of the earlier Asratian-Kuzjurin sampling-to-counting route, using the near-perfect-matching theorem of Gould and Kelly, sharpens the coefficient: for every fixed $k\\geq 3$, $\\sup_H \\overline{q}(H) \\leq (\\max\\{3,k-1\\}+o_d(1))\\log\\log d/\\log d$. Kahn's stronger pointwise prediction was disproved by Lee for $k\\geq 3$. The qualitative averaged conclusion was already recorded by Kahn and Kim, crediting Anders Johansson, and also follows from the Grable-Asratian-Kuzjurin enumeration; neither source states a rate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26133",
  "author_total": 1,
  "authors": [
    {
      "name": "Anish Gupta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6j01j929w117d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26133",
  "fulltext_url": null,
  "id": "6j01j929w117d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26133",
  "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": "W7214095010",
  "pdf_url": "https://arxiv.org/pdf/2609.26133",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6j01j929w117d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26133",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26133"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A rate for the average vacancy of uniformly random matchings in linear hypergraphs",
  "title_text": "A rate for the average vacancy of uniformly random matchings in linear hypergraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/6j01j929w117d",
  "venue": "arXiv",
  "versions": []
}