{
  "id": "2412.09532",
  "version": "v1",
  "published": "2024-12-12T18:21:33.000Z",
  "updated": "2024-12-12T18:21:33.000Z",
  "title": "Percolation on the stationary distributions of the voter model with stirring",
  "authors": [
    "Jhon Astoquillca",
    "Réka Szabó",
    "Daniel Valesin"
  ],
  "comment": "24 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "The voter model with stirring is a variant of the classical voter model with two possible opinions (0 and 1) that allows voters to interchange opinions at rate $\\mathsf{v} \\ge 0$. In [Astoquillca,24], it was proved that for this model on $\\mathbb Z^d$ for $d \\ge 3$ and any $\\mathsf{v}$, the set of extremal stationary measures is given by a family $\\{ \\mu_{\\alpha,\\mathsf{v}}: \\alpha \\in [0,1] \\}$, where $\\alpha$ is the density of voters with opinion 1. Sampling a configuration $\\xi$ from $\\mu_{\\alpha, \\mathsf v}$, we study $\\xi$ as a site percolation model on $\\mathbb Z^d$ where the set of occupied sites is the set of voters with opinion 1 in $\\xi$. Letting $\\alpha_c(\\mathsf v)$ be the supremum of all the values of $\\alpha$ for which percolation does not occur $\\mu_{\\alpha, \\mathsf v}$-a.s., we prove that $\\alpha_c(\\mathsf{v})$ converges to $p_c$, the critical density for classical Bernoulli site percolation, as $\\mathsf{v}$ tends to infinity. As a consequence, for $\\mathsf v$ large enough, the probability of the existence of an infinite percolation cluster of 1's exhibits a phase transition in $\\alpha$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-12T18:21:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82C22",
      "82B43"
    ],
    "keywords": [
      "stationary distributions",
      "extremal stationary measures",
      "classical bernoulli site percolation",
      "infinite percolation cluster",
      "site percolation model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}