{
  "id": "1912.06012",
  "version": "v1",
  "published": "2019-12-12T15:04:20.000Z",
  "updated": "2019-12-12T15:04:20.000Z",
  "title": "The phase transition for parking on Galton--Watson trees",
  "authors": [
    "Nicolas Curien",
    "Olivier Hénard"
  ],
  "comment": "15 pages, 5 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We establish a phase transition for the parking process on critical Galton--Watson trees. In this model, a random number of cars with mean $m$ and variance $\\sigma^{2}$ arrive independently on the vertices of a critical Galton--Watson tree with finite variance $\\Sigma^{2}$ conditioned to be large. The cars go down the tree and try to park on empty vertices as soon as possible. We show a phase transition depending on $$ \\Theta:= (1-m)^2- \\Sigma^2 (\\sigma^2+m^2-m).$$ Specifically, if $ \\Theta>0,$ then most cars will manage to park, whereas if $\\Theta<0$ then a positive fraction of the cars will not find a spot and exit the tree through the root. This confirms a conjecture of Goldschmidt and Przykucki.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-12T15:04:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05"
    ],
    "keywords": [
      "phase transition",
      "critical galton-watson tree",
      "random number",
      "finite variance",
      "empty vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}