{
  "id": "2208.04800",
  "version": "v1",
  "published": "2022-08-09T14:25:13.000Z",
  "updated": "2022-08-09T14:25:13.000Z",
  "title": "Distances in $\\frac{1}{|x-y|^{2d}}$ percolation models for all dimensions",
  "authors": [
    "Johannes Bäumler"
  ],
  "comment": "57 pages, 5 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study independent long-range percolation on $\\mathbb{Z}^d$ for all dimensions $d$, where the vertices $u$ and $v$ are connected with probability 1 for $\\|u-v\\|_\\infty=1$ and with probability $p(\\beta,\\{u,v\\})=1-e^{-\\beta \\int_{u+\\left[0,1\\right)^d} \\int_{v+\\left[0,1\\right)^d} \\frac{1}{\\|x-y\\|^{2d}} d x d y } \\approx \\frac{\\beta}{\\|u-v\\|^{2d}}$ for $\\|u-v\\|_\\infty \\geq 2$. Let $u \\in \\mathbb{Z}^d$ be a point with $\\|u\\|=n$. We show that both the graph distance between the origin $\\mathbf{0}$ and $u$ and the diameter of the box $\\{0 ,\\ldots, n\\}^d$ grow like $n^{\\theta(\\beta)}$, where $0<\\theta(\\beta ) < 1$. We also show that the graph distance has the same asymptotic growth when two vertices $u,v$ with $\\|u-v\\|_2 > 1$ are connected with a probability that is close enough to $p(\\beta,\\{u,v\\})$. Furthermore, we determine the asymptotic behavior of $\\theta(\\beta)$ for large $\\beta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-09T14:25:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "60K35",
      "82B27",
      "82B43"
    ],
    "keywords": [
      "percolation models",
      "dimensions",
      "study independent long-range percolation",
      "graph distance",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}