{
  "id": "1706.02597",
  "version": "v1",
  "published": "2017-06-08T14:11:03.000Z",
  "updated": "2017-06-08T14:11:03.000Z",
  "title": "Explosion and distances in scale-free percolation",
  "authors": [
    "Remco van der Hofstad",
    "Julia Komjathy"
  ],
  "comment": "46 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We investigate the weighted scale-free percolation (SFPW) model on $\\mathbb Z^d$. In the SFPW model, the vertices of $\\mathbb Z^d$ are assigned i.i.d. weights $(W_x)_{x\\in \\mathbb Z^d}$, following a power-law distribution with tail exponent $\\tau>1$. Conditioned on the collection of weights, the edges $(x,y)_{x, y \\in \\mathbb Z^d}$ are present independently with probability that a Poisson random variable with parameter $\\lambda W_x W_y/(\\|x-y\\|)^\\alpha$ is at least one, for some $\\alpha, \\lambda>0$, and where $\\|\\cdot \\|$ denotes the Euclidean distance. After the graph is constructed this way, we assign independent and identically distributed (i.i.d.) random edge lengths from distribution $L$ to all existing edges in the graph. The focus of the paper is to determine when is the obtained model \\emph{explosive}, that is, when it is possible to reach infinitely many vertices in finite time from a vertex. We show that explosion happens precisely for those edge length distributions that produce explosive branching processes with infinite mean power law offspring distributions. For non-explosive edge-length distributions, when $\\gamma \\in (1,2)$, we characterise the asymptotic behaviour of the time it takes to reach the first vertex that is graph distance $n$ away. For $\\gamma>2$, we show that the number of vertices reachable by time $t$ from the origin grows at most exponentially, thus explosion is never possible. For the non-explosive edge-length distributions, when $\\gamma \\in (1,2)$, we further determine the first order asymptotics of distances when $\\gamma\\in (1,2)$. As a corollary we obtain a sharp upper and lower bound for graph distances, closing a gap between a lower and upper bound on graph distances in Deijfen Hofstad `13 when $\\gamma \\in(1,2), \\tau>2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-08T14:11:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scale-free percolation",
      "graph distance",
      "non-explosive edge-length distributions",
      "mean power law offspring distributions",
      "edge length"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}