{
  "id": "1802.03813",
  "version": "v1",
  "published": "2018-02-11T20:44:07.000Z",
  "updated": "2018-02-11T20:44:07.000Z",
  "title": "Universality for 1d random band matrices: sigma-model approximation",
  "authors": [
    "Mariya Shcherbina",
    "Tatyana Shcherbina"
  ],
  "comment": "38 pp",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in J Stat Phys 164:1233 -- 1260, 2016; Commun Math Phys 351:1009 -- 1044, 2017. We consider random Hermitian block band matrices consisting of $W\\times W$ random Gaussian blocks (parametrized by $j,k \\in\\Lambda=[1,n]^d\\cap \\mathbb{Z}^d$) with a fixed entry's variance $J_{jk}=\\delta_{j,k}W^{-1}+\\beta\\Delta_{j,k}W^{-2}$, $\\beta>0$ in each block. Taking the limit $W\\to\\infty$ with fixed $n$ and $\\beta$, we derive the sigma-model approximation of the second correlation function similar to Efetov's one. Then, considering the limit $\\beta, n\\to\\infty$, we prove that in the dimension $d=1$ the behaviour of the sigma-model approximation in the bulk of the spectrum, as $\\beta\\gg n$, is determined by the classical Wigner -- Dyson statistics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-11T20:44:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "1d random band matrices",
      "sigma-model approximation",
      "supersymmetric transfer matrix approach",
      "block band matrices consisting",
      "random hermitian block band matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}