{
  "id": "1609.07043",
  "version": "v1",
  "published": "2016-09-22T16:00:15.000Z",
  "updated": "2016-09-22T16:00:15.000Z",
  "title": "On percolation critical probabilities and unimodular random graphs",
  "authors": [
    "Dorottya Beringer",
    "Gábor Pete",
    "Ádám Timár"
  ],
  "comment": "25 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We investigate generalisations of the classical percolation critical probabilities $p_c$, $p_T$ and the critical probability $\\tilde{p_c}$ defined by Duminil-Copin and Tassion (2015) to bounded degree unimodular random graphs. We further examine Schramm's conjecture in the case of unimodular random graphs: does $p_c(G_n)$ converge to $p_c(G)$ if $G_n\\to G$ in the local weak sense? Among our results are the following: 1. $p_c=\\tilde{p_c}$ holds for bounded degree unimodular graphs. However, there are unimodular graphs with sub-exponential volume growth and $p_T < p_c$; i.e., the classical sharpness of phase transition does not hold. 2. We give conditions which imply the inequality $\\liminf p_c(G_n) \\geq p_c(\\lim G_n)$. 3. There are sequences of unimodular graphs such that $G_n\\to G$ but $p_c(G)>\\lim p_c(G_n)$ or $p_c(G)<\\lim p_c(G_n)<1$. As a corollary to our positive results, we show that for any transitive graph with sub-exponential volume growth there is a sequence $T_n$ of large girth bi-Lipschitz invariant subgraphs such that $p_c(T_n)\\to 1$. It remains open whether this holds whenever the transitive graph has cost 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-22T16:00:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "percolation critical probabilities",
      "critical probability",
      "unimodular graphs",
      "sub-exponential volume growth",
      "large girth bi-lipschitz invariant subgraphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}