{
  "abstract": "Choose uniformly among the combinatorial types of convex three-dimensional polyhedra with a fixed admissible number $e$ of edges, and let $V_e$ be the number of vertices. This resolves a fixed-edge limit-distribution question posed by Rüdinger: if $β_e=V_e/(e+2)$, then $\\sqrt e(β_e-1/2)\\Rightarrow N(0,1/32)$, equivalently $\\operatorname{Var}(V_e)\\sim e/32$. Using the classical rooted enumeration and asymmetry results of Bender and Wormald, we derive a relative lattice local limit theorem on every $o(e^{3/4})$ window, a quartic correction from the rate function on every $o(e^{5/6})$ window, precise moderate-tail constants, a quadratic moderate-deviation principle for every $a_e\\to\\infty$ with $a_e=o(\\sqrt e)$, and a full speed-$e$ large-deviation principle with an explicit good rate function. All fixed standardised moments converge. When $e=3m$, the two extremal vertex counts have equal probability asymptotic to $\\frac{6561}{32\\sqrt2}(4/27)^m$. Rooted and unrooted fixed-edge laws are also uniformly exponentially close, with relative discrepancy $O(ρ^e)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22402",
  "author_total": 1,
  "authors": [
    {
      "name": "Lachlan Bridges"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5974pj7m5a7dn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22402",
  "fulltext_url": null,
  "id": "5974pj7m5a7dn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22402",
  "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.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214003550",
  "pdf_url": "https://arxiv.org/pdf/2609.22402",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5974pj7m5a7dn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22402",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22402"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Gaussian Vertex-Face Balance in Random Convex Polyhedra with Fixed Edge Count",
  "title_text": "Gaussian Vertex-Face Balance in Random Convex Polyhedra with Fixed Edge Count",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/5974pj7m5a7dn",
  "venue": "arXiv",
  "versions": []
}