{
  "id": "1905.01554",
  "version": "v1",
  "published": "2019-05-04T20:25:09.000Z",
  "updated": "2019-05-04T20:25:09.000Z",
  "title": "Fluctuation of the free energy of Sherrington-Kirkpatrick model with Curie-Weiss interaction: the paramagnetic regime",
  "authors": [
    "Debapratim Banerjee"
  ],
  "comment": "24 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider a spin system with pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by \\citet{Baiklee16} and \\citet{Baikleewu18} which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to \\citet{Baiklee16} in the \"paramagnetic regime\" when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the spherical symmetric case the free energy here can not be written as a function of the eigenvalues of the corresponding interaction matrix. The method in this paper relies on a dense sub-graph conditioning technique introduced by \\citet{Ban16}. The proof of the approximation by the linear spectral statistics part is taken from \\citet{Banerjee2017}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-04T20:25:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free energy",
      "curie-weiss interaction",
      "paramagnetic regime",
      "sherrington-kirkpatrick model",
      "fluctuation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}