{
  "id": "2104.08574",
  "version": "v1",
  "published": "2021-04-17T15:34:46.000Z",
  "updated": "2021-04-17T15:34:46.000Z",
  "title": "Hole probability for noninteracting fermions in a $d$-dimensional trap",
  "authors": [
    "Gabriel Gouraud",
    "Pierre Le Doussal",
    "Gregory Schehr"
  ],
  "comment": "Main text: 6 pages, 1 figure Supp mat: 15 pages, 6 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "The hole probability, i.e., the probability that a region is void of particles, is a benchmark of correlations in many body systems. We compute analytically this probability $P(R)$ for a spherical region of radius $R$ in the case of $N$ noninteracting fermions in their ground state in a $d$-dimensional trapping potential. Using a connection to the Laguerre-Wishart ensembles of random matrices, we show that, for large $N$ and in the bulk of the Fermi gas, $P(R)$ is described by a universal scaling function of $k_F R$, for which we obtain an exact formula ($k_F$ being the local Fermi wave-vector). It exhibits a super exponential tail $P(R)\\propto e^{- \\kappa_d (k_F R)^{d+1}}$ where $\\kappa_d$ is a universal amplitude, in good agreement with existing numerical simulations. When $R$ is of the order of the radius of the Fermi gas, the hole probability is described by a large deviation form which is not universal and which we compute exactly for the harmonic potential. Similar results also hold in momentum space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-17T15:34:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hole probability",
      "noninteracting fermions",
      "fermi gas",
      "large deviation form",
      "super exponential tail"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}