{
  "id": "1404.6238",
  "version": "v5",
  "published": "2014-04-24T19:34:45.000Z",
  "updated": "2015-06-04T06:04:09.000Z",
  "title": "Recurrence and transience for the frog model on trees",
  "authors": [
    "Christopher Hoffman",
    "Tobias Johnson",
    "Matthew Junge"
  ],
  "comment": "23 pages, 8 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite $d$-ary tree. We prove the model undergoes a phase transition, finding it recurrent for $d=2$ and transient for $d\\geq 5$. Simulations suggest strong recurrence for $d=2$, weak recurrence for $d=3$, and transience for $d\\geq 4$. Additionally, we prove a 0-1 law for all $d$-ary trees, and we exhibit a graph on which a 0-1 law does not hold.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-08-08T22:22:41.000Z",
      "abstract": "The frog model is a branching random walk that bifurcates on the first visit to each site. On the infinite rooted $d$-ary tree we prove a phase transition for recurrence and transience. When $d=2$ the root is visited infinitely many times and when $d\\geq 5$ the root is visited finitely many times. Additionally, we prove for all $d$ the model satisfies a 0-1 law for recurrence and transience and also exhibit a graph on which the frog model does not satisfy a 0-1 law. We conjecture that for $d=3$ the model remains recurrent, while for $d=4$ the model switches to being transient.",
      "comment": "21 pages, 8 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-06-04T06:04:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K10",
      "60J35"
    ],
    "keywords": [
      "frog model",
      "recurrence",
      "transience",
      "model remains recurrent",
      "ary tree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.6238H"
    }
  }
}