{
  "id": "1808.01863",
  "version": "v1",
  "published": "2018-08-06T13:02:26.000Z",
  "updated": "2018-08-06T13:02:26.000Z",
  "title": "The Contact Process on Periodic Trees",
  "authors": [
    "Remy Kassem",
    "Grayson York",
    "Brandon Zhao",
    "Xiangying Huang",
    "Rick Durrett"
  ],
  "comment": "26 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "A little over 25 years ago Pemantle pioneered the study of the contact process on trees, and showed that the critical values $\\lambda_1$ and $\\lambda_2$ for global and local survival were different. Here, we will consider the case of trees in which the degrees of vertices are periodic. We will compute bounds on $\\lambda_1$ and $\\lambda_2$ and for the corresponding critical values $\\lambda_g$ and $\\lambda_\\ell$ for branching random walk. Much of what we find for period two $(a,b)$ trees was known to Pemantle. However, two significant new results give sharp asymptotics for the critical value $\\lambda_2$ of $(1,n)$ trees and generalize that result to the $(a_1,\\ldots, a_k, n)$ tree when $\\max_i a_i \\le n^{1-\\epsilon}$ and $a_1 \\cdots a_k = n^b$. We also give results for $\\lambda_g$ and $\\lambda_\\ell$ on $(a,b,c)$ trees. Since the values come from solving cubic equations, the explicit formulas are not pretty, but it is surprising that they depend only on $a+b+c$ and $abc$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-06T13:02:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "contact process",
      "periodic trees",
      "critical value",
      "values come",
      "local survival"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}