{
  "id": "1306.0423",
  "version": "v2",
  "published": "2013-06-03T14:14:03.000Z",
  "updated": "2013-10-08T08:23:21.000Z",
  "title": "Typical versus averaged overlap distribution in Spin-Glasses : Evidence for the droplet scaling theory",
  "authors": [
    "Cecile Monthus",
    "Thomas Garel"
  ],
  "comment": "v2=final revised version (in particular new sections IIE,IIIC and Appendix B w.r.t. v1)",
  "journal": "Phys. Rev. B 88, 134204 (2013)",
  "doi": "10.1103/PhysRevB.88.134204",
  "categories": [
    "cond-mat.dis-nn"
  ],
  "abstract": "We consider the statistical properties over disordered samples of the overlap distribution $P_{\\cal J}(q)$ which plays the role of an order parameter in spin-glasses. We show that near zero temperature (i) the {\\it typical} overlap distribution is exponentially small in the central region of $-1<q<1$: $ P^{typ}(q) = e^{\\bar{\\ln P_{\\cal J}(q)}} \\sim e^{- \\beta N^{\\theta} \\phi(q)} $, where $\\theta$ is the droplet exponent defined here with respect to the total number $N$ of spins (in order to consider also fully connected models where the notion of length does not exist); (ii) the rescaled variable $v = - (\\ln P_{\\cal J}(q))/N^{\\theta}$ remains an O(1) random positive variable describing sample-to sample fluctuations; (iii) the averaged distribution $\\bar{P_{\\cal J}(q)} $ is non-typical and dominated by rare anomalous samples. Similar statements hold for the cumulative overlap distribution $I_{\\cal J}(q_0) \\equiv \\int_{0}^{q_0} dq P_{\\cal J}(q) $. These results are derived explicitly for the spherical mean-field model with $\\theta=1/3$, $\\phi(q)=1-q^2 $, and the random variable $v$ corresponds to the rescaled difference between the two largest eigenvalues of GOE random matrices. Then we compare numerically the typical and averaged overlap distributions for the long-ranged one-dimensional Ising spin-glass with random couplings decaying as $J(r) \\propto r^{-\\sigma}$ for various values of the exponent $\\sigma$, corresponding to various droplet exponents $\\theta(\\sigma)$, and for the mean-field SK-model (corresponding formally to the $\\sigma=0$ limit of the previous model). Our conclusion is that future studies on spin-glasses should measure the {\\it typical} values of the overlap distribution or of the cumulative overlap distribution to obtain clearer conclusions on the nature of the spin-glass phase.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-08T08:23:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "75.10.Nr",
      "05.50.+q",
      "64.60.-i",
      "75.40.Mg"
    ],
    "keywords": [
      "averaged overlap distribution",
      "droplet scaling theory",
      "describing sample-to sample fluctuations",
      "variable describing sample-to sample",
      "spin-glasses"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review B",
      "year": 2013,
      "month": "Oct",
      "volume": 88,
      "number": 13,
      "pages": 134204
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013PhRvB..88m4204M"
    }
  }
}