{
  "id": "2304.02898",
  "version": "v1",
  "published": "2023-04-06T07:11:10.000Z",
  "updated": "2023-04-06T07:11:10.000Z",
  "title": "Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere",
  "authors": [
    "Marcus Michelen",
    "Oren Yakir"
  ],
  "comment": "43 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CA",
    "math.MP"
  ],
  "abstract": "Smale's Seventh Problem asks for an efficient algorithm to generate a configuration of $n$ points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltr\\'an and Shub considered the set of points given by the stereographic projection of the roots of the random elliptic polynomial of degree $n$ and computed the expected logarithmic energy. We study the fluctuations of the logarithmic energy associated to this random configuration and prove a central limit theorem. Our approach shows that all cumulants of the logarithmic energy are asymptotically linear in $n$, and hence the energy is well-concentrated on the scale of $\\sqrt{n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-06T07:11:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random polynomials",
      "fluctuations",
      "smales seventh problem asks",
      "random elliptic polynomial",
      "central limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}