{
  "id": "2010.06736",
  "version": "v1",
  "published": "2020-10-13T23:27:09.000Z",
  "updated": "2020-10-13T23:27:09.000Z",
  "title": "Approximation on slabs and uniqueness for inhomogeneous percolation with a plane of defects",
  "authors": [
    "Bernardo N. B. de Lima",
    "Humberto C. Sanna",
    "Daniel Valesin"
  ],
  "comment": "29 pages, 6 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $ \\mathbb{L}^{d} = ( \\mathbb{Z}^{d},\\mathbb{E}^{d} ) $ be the $ d $-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on $ \\mathbb{L}^{d} $ in which every edge inside the $ s $-dimensional hyperplane $ \\mathbb{Z}^{s} \\times \\{ 0 \\}^{d-s} $, $ 2 \\leq s < d $, is open with probability $ q $ and every other edge is open with probability $ p $. We prove the uniqueness of the infinite cluster in the supercritical regime whenever $ p \\neq p_{c}(d) $, where $ p_{c}(d) $ denotes the threshold for homogeneous percolation, and that the critical point $ (p,q_{c}(p)) $ can be approximated on the phase space by the critical points of slabs, for any $ p < p_{c}(d) $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-13T23:27:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "inhomogeneous percolation",
      "uniqueness",
      "approximation",
      "dimensional hypercubic lattice",
      "critical point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}