{
  "id": "2205.02734",
  "version": "v1",
  "published": "2022-05-05T16:14:40.000Z",
  "updated": "2022-05-05T16:14:40.000Z",
  "title": "Percolation critical probabilities of matching lattice-pairs",
  "authors": [
    "Geoffrey R. Grimmett",
    "Zhongyang Li"
  ],
  "comment": "This is the second part of the previously posted article, available at arXiv:2203.00981, by the same authors, which has been split into two parts",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "A necessary and sufficient condition is established for the strict inequality $p_c(G_*)<p_c(G)$ between the critical probabilities of site percolation on a quasi-transitive, plane graph $G$ and on its matching graph $G_*$. It is assumed that $G$ is properly embedded in either the Euclidean or the hyperbolic plane. When $G$ is transitive, strict inequality holds if and only if $G$ is not a triangulation. The basic approach is the standard method of enhancements, but its implemention has complexity arising from the non-Euclidean (hyperbolic) space, the study of site (rather than bond) percolation, and the generality of the assumption of quasi-transitivity. This result is complementary to the work of the authors (\"Hyperbolic site percolation\", arXiv:2203.00981) on the equality $p_u(G) + p_c(G_*) = 1$, where $p_u$ is the critical probability for the existence of a unique infinite open cluster. More specifically, it implies for transitive $G$ that $p_u(G) + p_c(G) \\ge 1$, with equality if and only if $G$ is a triangulation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-05T16:14:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "percolation critical probabilities",
      "critical probability",
      "matching lattice-pairs",
      "unique infinite open cluster",
      "strict inequality holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}