{
  "id": "1711.08815",
  "version": "v1",
  "published": "2017-11-23T19:44:18.000Z",
  "updated": "2017-11-23T19:44:18.000Z",
  "title": "Positive association of the oriented percolation cluster in randomly oriented graphs",
  "authors": [
    "François Bienvenu"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Consider any fixed graph whose edges have been randomly and independently oriented, and write $\\{S \\leadsto i\\}$ to indicate that there is an oriented path going from a vertex $s \\in S$ to vertex $i$. Narayanan (2016) proved that for any set $S$ and any two vertices $i$ and $j$, $\\{S \\leadsto i\\}$ and $\\{S \\leadsto j\\}$ are positively correlated. His proof relies on the Ahlswede-Daykin inequality, a rather advanced tool of probabilistic combinatorics. In this short note, I give an elementary proof of the following, stronger result: writing $V$ for the vertex set of the graph, for any source set $S$, the events $\\{S \\leadsto i\\}$, $i \\in V$, are positively associated -- meaning that the expectation of the product of increasing functionals of the family $\\{S \\leadsto i\\}$ for $i \\in V$ is greater than the product of their expectations. To show how this result can be used in concrete calculations, I also detail the example of percolation from the leaves of the randomly oriented complete binary tree of height $n$. Positive association makes it possible to use the Stein--Chen method to find conditions for the size of the percolation cluster to be Poissonian in the limit as $n$ goes to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-23T19:44:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60K35",
      "82B43",
      "05C05"
    ],
    "keywords": [
      "oriented percolation cluster",
      "randomly oriented graphs",
      "positive association",
      "randomly oriented complete binary tree",
      "proof relies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}