{
  "abstract": "Let \\(d\\ge2\\) be fixed and let \\(G_n\\) be finite abelian groups of odd orders \\(N_n\\to\\infty\\). We determine the centered diameter-\\(d\\) critical window for the standard random Cayley graph in which each nonzero group element is selected independently. Writing \\(M_n=(N_n-1)/2\\), we prove that the normalized first distance-\\(d\\) coverage times satisfy \\sum_{[x]\\in(G_n\\setminus\\{0\\})/\\{\\pm1\\}} δ_{\\frac{N_n^{d-1}}{d!}τ_{n,[x]}^d-\\log M_n} \\xrightarrow{d} \\PPP(e^{-z} $\\,dz). Consequently, the number of antipodal defects in the critical window converges in total variation to a Poisson law, the diameter transition has the Gumbel profile \\(e^{-e^{-c}}\\), and the diameter hitting time has Gumbel fluctuations. In the original generator-density parametrization this yields the sharp fixed-\\(d\\) threshold constant \\(d!/2^d\\) throughout the odd-order abelian class. For \\(d=2\\), we additionally obtain an exact path--cycle decomposition of the target representation graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28573",
  "author_total": 1,
  "authors": [
    {
      "name": "Yao Zhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2sjzzqp5vxnae.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28573",
  "fulltext_url": null,
  "id": "2sjzzqp5vxnae",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28573",
  "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|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214280354",
  "pdf_url": "https://arxiv.org/pdf/2609.28573",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2sjzzqp5vxnae.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28573",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28573"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups",
  "title_text": "Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/2sjzzqp5vxnae",
  "venue": "arXiv",
  "versions": []
}