{
  "id": "1309.5661",
  "version": "v1",
  "published": "2013-09-22T22:33:38.000Z",
  "updated": "2013-09-22T22:33:38.000Z",
  "title": "Gap probabilities and applications to geometry and random topology",
  "authors": [
    "Antonio Lerario",
    "Erik Lundberg"
  ],
  "categories": [
    "math.PR",
    "math.AG",
    "math.AT",
    "math.DG"
  ],
  "abstract": "We give an exact formula for the value of the derivative at zero of the gap probability in finite n x n Gaussian ensembles. As n goes to infinity our computation provides an asymptotic (with an explicit constant) of the order n^(1/2). As a first application, we consider the set of n x n (Real, Complex or Quaternionic) Hermitian matrices with Frobenius norm one and determinant zero. We give an exact formula for the intrinsic volume of this set and as n goes to infinity its asymptotic (with an explicit constant) is of the order n^(1/2). As a second application we consider the problem of computing Betti numbers of an intersection of k random Kostlan quadrics in RP^n. We show that the i-th Betti number is asymptotically expected to be one (for i sufficiently away from n/2). In the case k=2 the the sum of all Betti numbers was recently shown by the first author to equal n+o(n). Here we sharpen this asymptotic proving an asymptotic with two orders of precision and explicit constants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-22T22:33:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "14P25",
      "53C65",
      "55R20"
    ],
    "keywords": [
      "gap probability",
      "random topology",
      "explicit constant",
      "application",
      "asymptotic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.5661L"
    }
  }
}