{
  "id": "1811.10249",
  "version": "v1",
  "published": "2018-11-26T09:37:26.000Z",
  "updated": "2018-11-26T09:37:26.000Z",
  "title": "The Coulomb gas, potential theory and phase transitions",
  "authors": [
    "Robert J. Berman"
  ],
  "comment": "40 pages",
  "categories": [
    "math-ph",
    "cond-mat.stat-mech",
    "math.CA",
    "math.CV",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We give a potential-theoretic characterization of measures which have the property that the corresponding Coulomb gas is \"well-behaved\" and similarly for more general Riesz gases. This means that the laws of the empirical measures of the corresponding random point process satisfy a Large Deviation Principle with a rate functional which depends continuously on the temperature, in the sense of Gamma-convergence. Equivalently, there is no zeroth-order phase transition at zero temperature. This is shown to be the case for the Hausdorff measure on a Lipschitz hypersurface. We also provide explicit examples of measures which are absolutely continuous with respect to Lesbesgue measure, such that the corresponding 2d Coulomb exhibits a zeroth-order phase transition. This is based on relations to Ullman's criterion in the theory of orthogonal polynomials and Bernstein-Markov inequalities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-26T09:37:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "potential theory",
      "zeroth-order phase transition",
      "corresponding random point process satisfy",
      "large deviation principle",
      "general riesz gases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}