{
  "id": "2207.13173",
  "version": "v1",
  "published": "2022-07-26T20:46:42.000Z",
  "updated": "2022-07-26T20:46:42.000Z",
  "title": "Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach",
  "authors": [
    "Philipp König",
    "Thomas Richthammer"
  ],
  "comment": "23 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "A layered graph $G^\\times$ is the Cartesian product of a graph $G = (V,E)$ with the linear graph $Z$, e.g. $Z^\\times$ is the 2D square lattice $Z^2$. For Bernoulli percolation with parameter $p \\in [0,1]$ on $G^\\times$ one intuitively would expect that $P_p((o,0) \\leftrightarrow (v,n)) \\ge P_p((o,0) \\leftrightarrow (v,n+1))$ for all $o,v \\in V$ and $n \\ge 0$. This is reminiscent of the better known bunkbed conjecture. Here we introduce an approach to the above monotonicity conjecture that makes use of a Markov chain building the percolation pattern layer by layer. In case of finite $G$ we thus can show that for some $N \\ge 0$ the above holds for all $n \\ge N$ $o,v \\in V$ and $p \\in [0,1]$. One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-26T20:46:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J10",
      "82B43",
      "15B51"
    ],
    "keywords": [
      "markov chain approach",
      "layered graph",
      "monotonicity properties",
      "problems concerning bernoulli percolation",
      "2d square lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}