{
  "id": "1810.07162",
  "version": "v1",
  "published": "2018-10-16T17:42:49.000Z",
  "updated": "2018-10-16T17:42:49.000Z",
  "title": "Critical probability on the product graph of a regular tree and a line",
  "authors": [
    "Kohei Yamamoto"
  ],
  "comment": "14 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider Bernoulli bond percolation on the product graph of a regular tree and a line. Schonmann showed that there are a.s. infinitely many infinite clusters at $p=p_u$ by using a certain function $\\alpha(p)$. The function $\\alpha(p)$ is defined by a exponential decay rate of probability that two vertices of the same layer are connected. We show the critical probability $p_c$ can be written by using $\\alpha(p)$. In other words, we construct another definition of the critical probability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-16T17:42:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical probability",
      "product graph",
      "regular tree",
      "bernoulli bond percolation",
      "exponential decay rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}