{
  "id": "2009.13337",
  "version": "v1",
  "published": "2020-09-28T14:03:24.000Z",
  "updated": "2020-09-28T14:03:24.000Z",
  "title": "An upper bound on the two-arms exponent for critical percolation on $\\mathbb{Z}^d$",
  "authors": [
    "Jacob van den Berg",
    "Diederik van Engelenburg"
  ],
  "comment": "10 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider critical site percolation on $\\mathbb{Z}^d$ with $d \\geq 2$. Cerf (2015) pointed out that from classical work by Aizenman, Kesten and Newman (1987) and Gandolfi, Grimmett and Russo (1988) one can obtain that the two-arms exponent is at least $1/2$. The paper by Cerf slightly improves that lower bound. Except for $d=2$ and for high $d$, no upper bound for this exponent seems to be known in the literature so far (not even implicity). We show that the distance-$n$ two-arms probability is at least $c n^{-(d^2 + 4 d -2)}$ (with $c >0$ a constant which depends on $d$), thus giving an upper bound $d^2 + 4 d -2$ for the above mentioned exponent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-28T14:03:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "upper bound",
      "two-arms exponent",
      "critical percolation",
      "critical site percolation",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}