{
  "id": "2501.02676",
  "version": "v1",
  "published": "2025-01-05T22:23:40.000Z",
  "updated": "2025-01-05T22:23:40.000Z",
  "title": "On the components of random geometric graphs in the dense limit",
  "authors": [
    "Mathew D. Penrose",
    "Xiaochuan Yang"
  ],
  "comment": "52 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the geometric graph on $n$ independent uniform random points in a connected compact region $A$ of ${\\bf R}^d, d \\geq 2$ with smooth boundary, or in the unit square, with distance parameter $r_n$. Let $K_n$ be the number of components of this graph, and $R_n$ the number of vertices not in the giant component. Let $S_n$ be the number of isolated vertices. We show that if $r_n$ is chosen so that $nr_n^d$ tends to infinity but slowly enough that ${\\bf E}[S_n]$ also tends to infinity, then $K_n$, $R_n$ and $S_n$ are all asymptotic to $\\mu_n$ in probability as $n \\to \\infty$ where (with $|A|$, $\\theta_d$ and $|\\partial A|$ denoting the volume of $A$, the volume of the unit $d$-ball, and the perimeter of $A$ respectively) $\\mu_n := ne^{-\\pi n r_n^d/|A|}$ if $d=2$ and $\\mu_n := ne^{-\\theta_d n r_n^d/|A|} + \\theta_{d-1}^{-1} |\\partial A| r_n^{1-d} e^{- \\theta_d n r_n^d/(2|A|)}$ if $d\\geq 3$. We also give variance asymptotics and central limit theorems for $K_n$ and $R_n$ in this limiting regime when $d \\geq 3$, and for Poisson input with $d \\geq 2$. We extend these results (substituting ${\\bf E}[S_n]$ for $\\mu_n$) to a class of non-uniform distributions on $A$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-05T22:23:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "60F05",
      "60F25",
      "05C80"
    ],
    "keywords": [
      "random geometric graphs",
      "dense limit",
      "independent uniform random points",
      "central limit theorems",
      "non-uniform distributions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}