{
  "id": "1109.3403",
  "version": "v2",
  "published": "2011-09-15T17:14:21.000Z",
  "updated": "2013-07-10T09:38:43.000Z",
  "title": "On the critical value function in the divide and color model",
  "authors": [
    "András Bálint",
    "Vincent Beffara",
    "Vincent Tassion"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "The divide and color model on a graph $G$ arises by first deleting each edge of $G$ with probability $1-p$ independently of each other, then coloring the resulting connected components (\\emph{i.e.}, every vertex in the component) black or white with respective probabilities $r$ and $1-r$, independently for different components. Viewing it as a (dependent) site percolation model, one can define the critical point $r_c^G(p)$. In this paper, we mainly study the continuity properties of the function $r_c^G$, which is an instance of the question of locality for percolation. Our main result is the fact that in the case $G=\\mathbb Z^2$, $r_c^G$ is continuous on the interval $[0,1/2)$; we also prove continuity at $p=0$ for the more general class of graphs with bounded degree. We then investigate the sharpness of the bounded degree condition and the monotonicity of $r_c^G(p)$ as a function of $p$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-10T09:38:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical value function",
      "color model",
      "bounded degree",
      "site percolation model",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.3403B"
    }
  }
}