{
  "id": "1509.04347",
  "version": "v1",
  "published": "2015-09-14T22:43:02.000Z",
  "updated": "2015-09-14T22:43:02.000Z",
  "title": "Maximally Persistent Cycles in Random Geometric Complexes",
  "authors": [
    "Omer Bobrowski",
    "Matthew Kahle",
    "Primoz Skraba"
  ],
  "categories": [
    "math.PR",
    "math.AT",
    "math.CO"
  ],
  "abstract": "We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree $k$ in persistent homology, for a either the \\v{C}ech or the Vietoris-Rips filtration built on a uniform Poisson process of intensity $n$ in the unit cube $[0,1]^d$. This is a natural way of measuring the largest \"$k$-dimensional hole\" in a random point set. This problem is in the intersection of geometric probability and algebraic topology, and is naturally motivated by a probabilistic view of topological inference. We show that for all $d \\ge 2$ and $1 \\le k \\le d-1$ the maximally persistent cycle has persistence of order $$ \\Theta \\left( \\left( \\frac{\\log n}{\\log \\log n} \\right)^{1/k} \\right),$$ with high probability, characterizing its rate of growth as $n \\to \\infty$. The implied constants depend on $k$, $d$, and on whether we consider the Vietoris-Rips or \\v{C}ech filtration.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-14T22:43:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B99",
      "60D05",
      "05E45",
      "55U10"
    ],
    "keywords": [
      "maximally persistent cycle",
      "random geometric complexes",
      "random geometric simplicial complexes",
      "persistent homology",
      "vietoris-rips filtration built"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150904347B"
    }
  }
}