{
  "id": "2308.06216",
  "version": "v1",
  "published": "2023-08-11T16:34:22.000Z",
  "updated": "2023-08-11T16:34:22.000Z",
  "title": "Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus",
  "authors": [
    "Bence Borda",
    "Peter Grabner",
    "Ryan W. Matzke"
  ],
  "comment": "24 pages",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so-called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere $\\mathbb{S}^d$ and the flat torus $\\mathbb{T}^d$, and the so-called spherical ensemble on $\\mathbb{S}^2$, which originates in random matrix theory. We extend results of Beltr\\'an, Marzo and Ortega-Cerd\\`a on the Riesz $s$-energy of the harmonic ensemble to the nonsingular regime $s<0$, and as a corollary find the expected value of the spherical cap $L^2$ discrepancy via the Stolarsky invariance principle. We also show that the spherical ensemble and the harmonic ensemble on $\\mathbb{S}^2$ and $\\mathbb{T}^2$ with $N$ points attain the optimal rate $N^{-1/2}$ in expectation in the Wasserstein metric $W_2$, in contrast to i.i.d. random points, which are known to lose a factor of $(\\log N)^{1/2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-11T16:34:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G55",
      "31C12",
      "49Q22",
      "11K38"
    ],
    "keywords": [
      "determinantal point processes",
      "flat torus",
      "optimal transport",
      "riesz energy",
      "discrepancy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}