{
  "id": "1307.2755",
  "version": "v1",
  "published": "2013-07-10T11:21:06.000Z",
  "updated": "2013-07-10T11:21:06.000Z",
  "title": "Confidence intervals for the critical value in the divide and color model",
  "authors": [
    "András Bálint",
    "Vincent Beffara",
    "Vincent Tassion"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We obtain confidence intervals for the location of the percolation phase transition in H\\\"aggstr\\\"om's divide and color model on the square lattice $\\mathbb{Z}^2$ and the hexagonal lattice $\\mathbb{H}$. The resulting probabilistic bounds are much tighter than the best deterministic bounds up to date; they give a clear picture of the behavior of the DaC models on $\\mathbb{Z}^2$ and $\\mathbb{H}$ and enable a comparison with the triangular lattice $\\mathbb{T}$. In particular, our numerical results suggest similarities between DaC model on these three lattices that are in line with universality considerations, but with a remarkable difference: while the critical value function $r_c(p)$ is known to be constant in the parameter $p$ for $p<p_c$ on $\\mathbb{T}$ and appears to be linear on $\\mathbb{Z}^2$, it is almost certainly non-linear on $\\mathbb{H}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-10T11:21:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "color model",
      "confidence intervals",
      "dac model",
      "percolation phase transition",
      "best deterministic bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.2755B"
    }
  }
}