{
  "id": "1901.04906",
  "version": "v1",
  "published": "2019-01-15T16:16:58.000Z",
  "updated": "2019-01-15T16:16:58.000Z",
  "title": "Cover time for branching random walks on regular trees",
  "authors": [
    "Matthew I. Roberts"
  ],
  "comment": "15 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $T$ be the regular tree in which every vertex has exactly $d\\ge 3$ neighbours. Run a branching random walk on $T$, in which at each time step every particle gives birth to a random number of children with mean $d$ and finite variance, and each of these children moves independently to a uniformly chosen neighbour of its parent. We show that, starting with one particle at some vertex $0$ and conditionally on survival of the process, the time it takes for every vertex within distance $r$ of $0$ to be hit by a particle of the branching random walk is $r + \\frac{1}{\\log(3/2)}\\log\\log r + o(\\log\\log r)$. As part of the proof we develop apparently new bounds on the distribution of the total progeny up to generation $n$ of a critical Galton-Watson process. These we prove using a simple minimal inequality for supermartingales.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-15T16:16:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "60G42",
      "60D05"
    ],
    "keywords": [
      "branching random walk",
      "regular tree",
      "cover time",
      "simple minimal inequality",
      "uniformly chosen neighbour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}