{
  "id": "1504.02731",
  "version": "v1",
  "published": "2015-04-10T16:20:02.000Z",
  "updated": "2015-04-10T16:20:02.000Z",
  "title": "Some Properties of the Phase Diagram for Mixed $p$-Spin Glasses",
  "authors": [
    "Aukosh Jagannath",
    "Ian Tobasco"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we study the Parisi variational problem for mixed $p$-spin glasses with Ising spins. We present a new characterization of Parisi measures whose origin lies in the first order optimality conditions for the Parisi functional, which is known to be strictly convex. Using this characterization, we study the phase diagram in the temperature-external field plane. We begin by deriving self-consistency conditions for Parisi measures that generalize those of de Almeida and Thouless to all levels of Replica Symmetry Breaking (RSB) and all models. As a consequence, we conjecture that for all models the Replica Symmetric (RS) phase is the region determined by the natural analogue of the de Almeida-Thouless condition. We show that for all models, the complement of this region is in the RSB phase. Furthermore, we show that the conjectured phase boundary is exactly the phase boundary at sufficiently low temperature and positive external field. In the case of the Sherrington-Kirkpatrick model, we prove that the de Almeida-Thouless line is the phase boundary in the plane less a bounded set that does not contain the critical point at zero external field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-10T16:20:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B44",
      "82D30",
      "49S05",
      "49K21"
    ],
    "keywords": [
      "spin glasses",
      "phase diagram",
      "phase boundary",
      "properties",
      "parisi measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150402731J"
    }
  }
}