{
  "id": "1909.10441",
  "version": "v1",
  "published": "2019-09-23T15:51:43.000Z",
  "updated": "2019-09-23T15:51:43.000Z",
  "title": "The Contact Process on Periodic Trees",
  "authors": [
    "Xiangying Huang",
    "Rick Durrett"
  ],
  "comment": "12 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "A little over 25 years ago Pemantle pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values $\\lambda_1$ and $\\lambda_2$ for global and local survival were different. He also considered trees with periodic degree sequences, and Galton-Watson trees. Here, we will consider periodic trees in which the number of children in successive generation is $(n,a_1,\\ldots, a_k)$ with $\\max_i a_i \\le Cn^{1-\\delta}$ and $\\log(a_1 \\cdots a_k)/\\log n \\to b$ as $n\\to\\infty$. We show that the critical value for local survival is asymptotically $\\sqrt{c (\\log n)/n}$ where $c=(k-b)/2$. This supports Pemantle's claim that the critical value is largely determined by the maximum degree, but it also shows that the smaller degrees can make a significant contribution to the answer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-23T15:51:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "periodic trees",
      "contact process",
      "critical value",
      "local survival",
      "supports pemantles claim"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}