{
  "id": "2405.09343",
  "version": "v1",
  "published": "2024-05-15T13:46:26.000Z",
  "updated": "2024-05-15T13:46:26.000Z",
  "title": "On the number of infinite clusters in the constrained-degree percolation model",
  "authors": [
    "Weberson S. Arcanjo",
    "Alan S. Pereira",
    "Diogo C. dos Santos"
  ],
  "comment": "15 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider constrained-degree percolation model on the hypercubic lattice \\linebreak $\\mathbb{L}^d=(\\mathbb{Z}^d,\\mathbb{E}^d)$. In this model, there exits a sequence $(U_e)_{e\\in\\mathbb{E}^d}$ of i.i.d. random variables with distribution $Unif([0,1])$ and a positive integer $k$, which is called a constraint. Each edge $e$ attempts to open at time $U_e$, and the attempt is successful if the number of neighboring edges open at each endvertex of $e$ is at most $k-1$. In \\cite{hartarsky2022weakly}, the authors demonstrated that this model undergoes a phase transition when $d\\geq3$ and for most nontrivial values of $k$. In the present work, we prove that, for any fixed constraint, the number of infinite clusters at any given time $t\\in[0,1)$ is either 0 or 1, almost surely.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-15T13:46:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "constrained-degree percolation model",
      "infinite clusters",
      "random variables",
      "constraint",
      "hypercubic lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}