{
  "id": "1910.12803",
  "version": "v1",
  "published": "2019-10-28T16:58:05.000Z",
  "updated": "2019-10-28T16:58:05.000Z",
  "title": "Homotopy Types of Random Cubical Complexes",
  "authors": [
    "Kenneth Dowling",
    "Erik Lundberg"
  ],
  "comment": "21 pages, 5 figures, 2 tables",
  "categories": [
    "math.PR",
    "math.AT",
    "math.CO"
  ],
  "abstract": "We study the topology of a random cubical complex associated to Bernoulli site percolation on a cubical grid. We begin by establishing a limit law for homotopy types. More precisely, looking within an expanding window, we define a sequence of normalized counting measures (counting connected components according to homotopy type), and we show that this sequence of random probability measures converges in probability to a deterministic probability measure. We then investigate the dependence of the limiting homotopy measure on the coloring probability $p$, and our results show a qualitative change in the homotopy measure as $p$ crosses the percolation threshold $p=p_c$. Specializing to the case of $d=2$ dimensions, we also present empirical results that raise further questions on the $p$-dependence of the limiting homotopy measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-28T16:58:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "55U10",
      "05C80"
    ],
    "keywords": [
      "random cubical complex",
      "homotopy type",
      "limiting homotopy measure",
      "random probability measures converges",
      "deterministic probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}