{
  "abstract": "Let $G(n,p,m)$ be the random subgraph of the cycle power $C_n^m$ obtained by retaining each edge independently with probability $p$. We study connectivity, component geometry, and random walks throughout the regime $\\log n\\ll m<n/2$. We prove that $p\\sim(\\log n)/(2m)$ is a sharp threshold for connectivity. If $n(1-p)^{2m}\\toλ\\in(0,\\infty)$, then the number of isolated vertices converges to $\\mathrm{Poisson}(λ)$, and with high probability every other vertex lies in a single component $C_1$. In particular, the connectivity probability converges to $e^{-λ}$. The component $C_1$ contains a cyclic chain of connected sets occupying all but an $o(1)$ fraction of each consecutive block of length $\\asymp m$, with $Θ(m\\log n)$ edges between consecutive sets. Its diameter has order $n/m+\\log n/\\log\\log n$ with high probability. For every fixed $c>1$, uniformly over $p\\geq c\\log n/(2m)$, the lazy-walk spectral gap is asymptotic in probability to that of $C_n^m$. In the same range, the mean hitting time of simple random walk satisfies, uniformly over distinct $x,y$, $$ H_G(x,y)=\\frac{2|E(G)|}{d_G(y)} +\\frac{r_{xy}(n-r_{xy})}{κ_m} +o_{\\mathrm{Prob}}\\left(n+\\frac{n^2}{m^2}\\right), $$ where $r_{xy}$ is cyclic distance between vertices $x$ and $y$, $d_G(z)$ is the degree of vertex $z$ and $κ_m=(m+1)(2m+1)/6$. Thus, for this random graph model, connectivity has the Erdős--Rényi asymptotics with $2m$ possible neighbours, while distances, relaxation, and hitting times retain the geometry of the underlying cycle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.42107",
  "author_total": 3,
  "authors": [
    {
      "name": "Shantanav Chakraborty"
    },
    {
      "name": "Kyle Luh"
    },
    {
      "name": "Rudransh Pratap Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6d1dpmjy18f5y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.42107",
  "fulltext_url": null,
  "id": "6d1dpmjy18f5y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.42107",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.42107",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6d1dpmjy18f5y.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.42107",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.42107"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Band Random Graphs: Connectivity, Geometry, and Transport",
  "title_text": "Band Random Graphs: Connectivity, Geometry, and Transport",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/6d1dpmjy18f5y",
  "venue": "arXiv",
  "versions": []
}