{
  "id": "2206.13695",
  "version": "v1",
  "published": "2022-06-28T02:00:04.000Z",
  "updated": "2022-06-28T02:00:04.000Z",
  "title": "Critical parameter of the frog model on homogeneous trees with geometric lifetime",
  "authors": [
    "Sandro Gallo",
    "Caio Pena"
  ],
  "comment": "12 pages 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the frog model with geometric lifetime (parameter $1-p$) on homogeneous trees of dimension $d$. In 2002, \\cite{alves2002-2} proved that there exists a critical lifetime parameter $p_c\\in(0,1)$ above which infinitely many frogs are activated with positive probability, and they gave lower and upper bounds for $p_c$. Since then, the literature on this model focussed on refinements of the upper bound. In the present paper we improve the bounds for $p_c$ \\emph{on both sides}. We also provide a discussion comparing the bounds of the literature and their proofs. Our proofs are based on coupling.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-28T02:00:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "05C81",
      "60K05"
    ],
    "keywords": [
      "frog model",
      "geometric lifetime",
      "homogeneous trees",
      "critical parameter",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}