{
  "id": "1502.03051",
  "version": "v1",
  "published": "2015-02-10T19:49:03.000Z",
  "updated": "2015-02-10T19:49:03.000Z",
  "title": "A new proof of the sharpness of the phase transition for Bernoulli percolation on $\\mathbb Z^d$",
  "authors": [
    "Hugo Duminil-Copin",
    "Vincent Tassion"
  ],
  "comment": "6 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that - for $p<p_c$, the probability that the origin is connected by an open path to distance $n$ decays exponentially fast in $n$. - for $p>p_c$, the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound $\\theta(p)\\ge\\tfrac{p-p_c}{p(1-p_c)}$. This note presents the argument of \\cite{DumTas15}, which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework of nearest-neighbour Bernoulli percolation on $\\mathbb Z^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-10T19:49:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "phase transition",
      "nearest-neighbour bernoulli percolation",
      "long-range bernoulli percolation",
      "mean-field lower bound",
      "infinite cluster satisfies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}