{
  "id": "1810.06040",
  "version": "v1",
  "published": "2018-10-14T14:40:52.000Z",
  "updated": "2018-10-14T14:40:52.000Z",
  "title": "The Contact Process on Random Graphs and Galton-Watson Trees",
  "authors": [
    "Xiangying Huang",
    "Rick Durrett"
  ],
  "comment": "23 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival $\\lambda_2=0$ and (ii) when it is geometric($p$) we have $\\lambda_2 \\le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $\\lambda_c(n)$ for \"prolonged persistence\" on graphs with $n$ vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known $\\lambda_c(n) \\to 0$ we give estimates on the rate of convergence. Physicists tell us that $\\lambda_c(n) \\sim 1/\\Lambda(n)$ where $\\Lambda(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-14T14:40:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "contact process",
      "galton-watson trees",
      "random graphs",
      "critical value",
      "maximum eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}