{
  "id": "1811.05495",
  "version": "v1",
  "published": "2018-11-13T19:02:47.000Z",
  "updated": "2018-11-13T19:02:47.000Z",
  "title": "Phase transition for the frog model on biregular trees",
  "authors": [
    "Elcio Lebensztayn",
    "Jaime Utria"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the frog model with death on the biregular tree $\\mathbb{T}_{d_1,d_2}$. Initially, there is a random number of awake and sleeping particles located on the vertices of the tree. Each awake particle moves as a discrete-time independent simple random walk on $\\mathbb{T}_{d_1,d_2}$ and has a probability of death $(1-p)$ before each step. When an awake particle visits a vertex which has not been visited previously, the sleeping particles placed there are awakened. We prove that this model undergoes a phase transition: for values of $p$ below a critical probability $p_c$, the system dies out almost surely, and for $p > p_c$, the system survives with positive probability. We establish explicit bounds for $p_c$ in the case of random initial configuration. For the model starting with one particle per vertex, the critical probability satisfies $p_c(\\mathbb{T}_{d_1,d_2}) = 1/2 + \\Theta(1/d_1+1/d_2)$ as $d_1, d_2 \\to \\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-13T19:02:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "frog model",
      "phase transition",
      "biregular tree",
      "discrete-time independent simple random walk",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}