{
  "id": "2505.09025",
  "version": "v1",
  "published": "2025-05-13T23:43:49.000Z",
  "updated": "2025-05-13T23:43:49.000Z",
  "title": "Mean-field behaviour of the random connection model on hyperbolic space",
  "authors": [
    "Matthew Dickson",
    "Markus Heydenreich"
  ],
  "comment": "37 pages, 6 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the random connection model on hyperbolic space $\\mathbb{H}^d$ in dimension $d=2,3$. Vertices of the spatial random graph are given as a Poisson point process with intensity $\\lambda>0$. Upon variation of $\\lambda$ there is a percolation phase transition: there exists a critical value $\\lambda_c>0$ such that for $\\lambda<\\lambda_c$ all clusters are finite, but infinite clusters exist for $\\lambda>\\lambda_c$. We identify certain critical exponents that characterize the clusters at (and near) $\\lambda_c$, and show that they agree with the mean-field values for percolation. We derive the exponents through isoperimetric properties of critical percolation clusters rather than via a calculation of the triangle diagram.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-13T23:43:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43",
      "60G55"
    ],
    "keywords": [
      "random connection model",
      "hyperbolic space",
      "mean-field behaviour",
      "percolation phase transition",
      "poisson point process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}