{
  "id": "2104.09408",
  "version": "v1",
  "published": "2021-04-19T15:57:32.000Z",
  "updated": "2021-04-19T15:57:32.000Z",
  "title": "Number-Rigidity and $β$-Circular Riesz gas",
  "authors": [
    "David Dereudre",
    "Thibaut Vasseur"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "For an inverse temperature $\\beta>0$, we define the $\\beta$-circular Riesz gas on $\\mathbb{R}^d$ as any microscopic thermodynamic limit of Gibbs particle systems on the torus interacting via the Riesz potential $g(x) = \\Vert x \\Vert^{-s}$. We focus on the non integrable case $d-1<s<d$. Our main result ensures, for any dimension $d\\ge 1$ and inverse temperature $\\beta>0$, the existence of a $\\beta$-circular Riesz gas which is not number-rigid. Recall that a point process is said number rigid if the number of points in a bounded Borel set $\\Delta$ is a function of the point configuration outside $\\Delta$. It is the first time that the non number-rigidity is proved for a Gibbs point process interacting via a non integrable potential. We follow a statistical physics approach based on the canonical DLR equations. It is inspired by Dereudre-Hardy-Lebl\\'e and Ma\\\"ida (2021) where the authors prove the number-rigidity of the $\\text{Sine}_\\beta$ process.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-19T15:57:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "circular riesz gas",
      "number-rigidity",
      "inverse temperature",
      "microscopic thermodynamic limit",
      "gibbs point process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}