{
  "id": "1909.06913",
  "version": "v1",
  "published": "2019-09-15T23:47:31.000Z",
  "updated": "2019-09-15T23:47:31.000Z",
  "title": "Periodic solutions of one-dimensional cellular automata with random rules",
  "authors": [
    "Janko Gravner",
    "Xiaochen Liu"
  ],
  "categories": [
    "math.PR",
    "math.DS",
    "nlin.CG"
  ],
  "abstract": "We study cellular automata with randomly selected rules. Our setting are two-neighbor rules with a large number $n$ of states. The main quantity we analyze is the asymptotic probability, as $n \\to \\infty$, that the random rule has a periodic solution with given spatial and temporal periods. We prove that this limiting probability is non-trivial when the spatial and temporal periods are confined to a finite range. The main tool we use is the Chen-Stein method for Poisson approximation. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-15T23:47:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "37B15",
      "68Q80"
    ],
    "keywords": [
      "one-dimensional cellular automata",
      "periodic solution",
      "random rule",
      "study cellular automata",
      "limiting probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}