{
  "id": "1502.01306",
  "version": "v1",
  "published": "2015-02-04T19:43:17.000Z",
  "updated": "2015-02-04T19:43:17.000Z",
  "title": "Percolation on the stationary distributions of the voter model",
  "authors": [
    "Balazs Rath",
    "Daniel Valesin"
  ],
  "comment": "35 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "The voter model on $\\mathbb{Z}^d$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When $d \\geq 3$, the set of (extremal) stationary distributions is a family of measures $\\mu_\\alpha$, for $\\alpha$ between 0 and 1. A configuration sampled from $\\mu_\\alpha$ is a strongly correlated field of 0's and 1's on $\\mathbb{Z}^d$ in which the density of 1's is $\\alpha$. We consider such a configuration as a site percolation model on $\\mathbb{Z}^d$. We prove that if $d \\geq 5$, the probability of existence of an infinite percolation cluster of 1's exhibits a phase transition in $\\alpha$. If the voter model is allowed to have sufficiently spread-out interactions, we prove the same result for $d \\geq 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-04T19:43:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "voter model",
      "stationary distributions",
      "site percolation model",
      "infinite percolation cluster",
      "particle system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150201306R"
    }
  }
}