{
  "id": "0912.2875",
  "version": "v2",
  "published": "2009-12-15T13:07:30.000Z",
  "updated": "2010-02-25T15:37:25.000Z",
  "title": "Matching between typical fluctuations and large deviations in disordered systems : application to the statistics of the ground state energy in the SK spin-glass model",
  "authors": [
    "Cecile Monthus",
    "Thomas Garel"
  ],
  "comment": "10 pages, 4 figures",
  "journal": "J. Stat. Mech. (2010) P02023",
  "doi": "10.1088/1742-5468/2010/02/P02023",
  "categories": [
    "cond-mat.dis-nn"
  ],
  "abstract": "For the statistics of global observables in disordered systems, we discuss the matching between typical fluctuations and large deviations. We focus on the statistics of the ground state energy $E_0$ in two types of disordered models : (i) for the directed polymer of length $N$ in a two-dimensional medium, where many exact results exist (ii) for the Sherrington-Kirkpatrick spin-glass model of $N$ spins, where various possibilities have been proposed. Here we stress that, besides the behavior of the disorder-average $E_0^{av}(N)$ and of the standard deviation $ \\Delta E_0(N) \\sim N^{\\omega_f}$ that defines the fluctuation exponent $\\omega_f$, it is very instructive to study the full probability distribution $\\Pi(u)$ of the rescaled variable $u= \\frac{E_0(N)-E_0^{av}(N)}{\\Delta E_0(N)}$ : (a) numerically, the convergence towards $\\Pi(u)$ is usually very rapid, so that data on rather small sizes but with high statistics allow to measure the two tails exponents $\\eta_{\\pm}$ defined as $\\ln \\Pi(u \\to \\pm \\infty) \\sim - | u |^{\\eta_{\\pm}}$. In the generic case $1< \\eta_{\\pm} < +\\infty$, this leads to explicit non-trivial terms in the asymptotic behaviors of the moments $\\bar{Z_N^n}$ of the partition function when the combination $[| n | N^{\\omega_f}]$ becomes large (b) simple rare events arguments can usually be found to obtain explicit relations between $\\eta_{\\pm}$ and $\\omega_f$. These rare events usually correspond to 'anomalous' large deviation properties of the generalized form $R(w_{\\pm} = \\frac{E_0(N)-E_0^{av}(N)}{N^{\\kappa_{\\pm}}}) \\sim e^{- N^{\\rho_{\\pm}} {\\cal R}_{\\pm}(w_{\\pm})}$ (the 'usual' large deviations formalism corresponds to $\\kappa_{\\pm}=1=\\rho_{\\pm}$).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-02-25T15:37:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ground state energy",
      "sk spin-glass model",
      "typical fluctuations",
      "disordered systems",
      "statistics"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Statistical Mechanics: Theory and Experiment",
      "year": 2010,
      "month": "Feb",
      "volume": 2010,
      "number": 2,
      "pages": 2023
    },
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010JSMTE..02..023M"
    }
  }
}