{
  "abstract": "Consider the number of faces arising from unimodular triangulations of random polytopes associated with Erdős--Rényi graphs. Focusing on two-dimensional faces of regular unimodular triangulations of random symmetric edge polytopes, we derive precise asymptotic expansions for the expectation and variance and then establish a quantitative central limit theorem with an explicit Berry--Esseen bound. The proof combines an algebraically derived characterization of faces in terms of local graph configurations with a detailed combinatorial--geometric study of the relevant configurations. The normal approximation is established using the discrete Malliavin--Stein method. We also determine the asymptotic orders of the expectation and variance and establish a quantitative central limit theorem for the number of faces in each fixed higher dimension. These results identify a common local mechanism underlying the fluctuations of face numbers and extend the distributional theory of random symmetric edge polytopes [Donzelmann et al., arXiv:2603.10650] beyond edge statistics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28088",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhonggen Su"
    },
    {
      "name": "Huilian Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vbex0pcgdd81.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28088",
  "fulltext_url": null,
  "id": "3vbex0pcgdd81",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28088",
  "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": "W7223097969",
  "pdf_url": "https://arxiv.org/pdf/2610.28088",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vbex0pcgdd81.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28088",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28088"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Berry--Esseen Bounds for Faces in Unimodular Triangulations of Random Polytopes",
  "title_text": "Berry--Esseen Bounds for Faces in Unimodular Triangulations of Random Polytopes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/3vbex0pcgdd81",
  "venue": "arXiv",
  "versions": []
}