{
  "id": "1609.08738",
  "version": "v1",
  "published": "2016-09-28T02:30:50.000Z",
  "updated": "2016-09-28T02:30:50.000Z",
  "title": "Frogs on trees ?",
  "authors": [
    "Jonathan Hermon"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a system of simple random walks on $\\mathcal{T}_{d,n} = \\mathcal{V}_{d,n}, \\mathcal{E}_{d,n})$, the $d$-ary tree of depth $n$, known as the frog model. Initially there are Pois($\\lambda$) particles at each site, independently, with one additional particle planted at some vertex $\\mathbf{o}$. Initially all particles are inactive, except for the ones which are placed at $\\mathbf{o}$. Active particles perform (independent) $ t \\in \\mathbb{N} \\cup \\{\\infty \\} $ steps of simple random walk on the tree. When an active particle hits an inactive particle, the latter becomes active. The model is often interpreted as a model for a spread of an epidemic. As such, it is natural to investigate whether the entire population is eventually infected, and if so, how quickly does this happen. Let $\\mathcal{R}_t$ be the set of vertices which are visited by the process. Let $\\mathcal{S}(\\mathcal{T}_{d,n}) := \\inf \\{t:\\mathcal{R}_t = \\mathcal{V}_{d,n} \\} $. Let $\\mathrm{CT}(\\mathcal{T}_{d,n})$ be first time in which every vertex was visited at least once, when we take $t=\\infty$. We show that there exist absolute constants, $c,C>0$ such that for all fixed $\\lambda >0$, w.h.p.~$c \\le \\lambda \\mathcal{S}(\\mathcal{T}_{d,n}) /n \\log n \\le C$ and $\\mathrm{CT}(\\mathcal{T}_{d,n}) \\le 2^{C\\sqrt{ \\log |\\mathcal{V}_{d,n}| }}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-28T02:30:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "05C81"
    ],
    "keywords": [
      "simple random walk",
      "additional particle",
      "absolute constants",
      "active particles perform",
      "frog model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}