{
  "id": "2209.00396",
  "version": "v1",
  "published": "2022-09-01T12:19:49.000Z",
  "updated": "2022-09-01T12:19:49.000Z",
  "title": "Decay of correlations and thermodynamic limit for the circular Riesz gas",
  "authors": [
    "Jeanne Boursier"
  ],
  "comment": "Preliminary version, comments welcome",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We investigate the thermodynamic limit of the circular long-range Riesz gas, a system of particles interacting pairwise through an inverse power kernel. We show that after rescaling, so that the typical spacing of particles is of order $1$, the microscopic point process converges as the number of points tends to infinity, to an infinite volume measure $\\mathrm{Riesz}_{s,\\beta}$. This convergence result is obtained by analyzing gaps correlations, which are shown to decay in power-law with exponent $2-s$. One also proves that the decay of correlations is much faster for the hypersingular Riesz gas, thereby exhibiting a discontinuous transition at $s=1$. Our method is based on the analysis of the Helffer-Sj\\\"ostrand equation in its static form and on various discrete elliptic regularity estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-01T12:19:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "circular riesz gas",
      "thermodynamic limit",
      "correlations",
      "microscopic point process converges",
      "circular long-range riesz gas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}