{
  "id": "1910.02367",
  "version": "v1",
  "published": "2019-10-06T04:23:33.000Z",
  "updated": "2019-10-06T04:23:33.000Z",
  "title": "The frog model on Galton-Watson trees",
  "authors": [
    "Marcus Michelen",
    "Josh Rosenberg"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider an interacting particle system on trees known as the frog model: initially, a single active particle begins at the root and i.i.d. $\\mathrm{Poiss}(\\lambda)$ many inactive particles are placed at each non-root vertex. Active particles perform discrete time simple random walk and activate the inactive particles they encounter. We show that for Galton-Watson trees with offspring distributions $Z$ satisfying $\\mathbf{P}(Z \\geq 2) = 1$ and $\\mathbf{E}[Z^{4 + \\epsilon}] < \\infty$ for some $\\epsilon > 0$, there is a critical value $\\lambda_c\\in(0,\\infty)$ separating recurrent and transient regimes for almost surely every tree, thereby answering a question of Hoffman-Johnson-Junge. In addition, we also establish that this critical parameter depends on the entire offspring distribution, not just the maximum value of $Z$, answering another question of Hoffman-Johnson-Junge and showing that the frog model and contact process behave differently on Galton-Watson trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-06T04:23:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "galton-watson trees",
      "frog model",
      "particles perform discrete time simple",
      "discrete time simple random walk",
      "perform discrete time simple random"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}