{
  "id": "1710.05884",
  "version": "v1",
  "published": "2017-10-16T17:29:44.000Z",
  "updated": "2017-10-16T17:29:44.000Z",
  "title": "Infection spread for the frog model on trees",
  "authors": [
    "Christopher Hoffman",
    "Tobias Johnson",
    "Matthew Junge"
  ],
  "comment": "44 pages plus appendice; 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the frog model with initial density of particles $\\mu$. On the infinite $d$-ary tree for $\\mu = \\Omega(d^2)$, we show that the set of activated sites contains a linearly expanding ball. This helps us deduce that on the full $d$-ary tree of height $n$,it takes $O(n\\log n)$ steps to visit all sites of the tree with high probability. Conversely, a different argument shows that it takes $\\exp(\\Omega(\\sqrt{n}))$ steps if $\\mu=O(d)$. Both bounds are sharp. It was previously unknown whether the cover time was polynomial or superpolynomial for any value of $\\mu$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-16T17:29:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J80",
      "60J10"
    ],
    "keywords": [
      "frog model",
      "infection spread",
      "ary tree",
      "activated sites contains",
      "initial density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}