{
  "id": "1311.0395",
  "version": "v1",
  "published": "2013-11-02T17:21:59.000Z",
  "updated": "2013-11-02T17:21:59.000Z",
  "title": "Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails",
  "authors": [
    "Marek Biskup",
    "Wolfgang Koenig"
  ],
  "comment": "36 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider random Schr\\\"odinger operators of the form $\\Delta+\\xi$, where $\\Delta$ is the lattice Laplacian on $\\mathbb Z^d$ and $\\xi$ is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of $\\mathbb Z^d$. We show that for $\\xi$ with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where $\\xi$ takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-02T17:21:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H25",
      "82B44",
      "82C44",
      "60J80"
    ],
    "keywords": [
      "random schrödinger operators",
      "eigenvalue order statistics",
      "doubly-exponential tails",
      "upper extreme order statistics",
      "gumbel max-order class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.0395B"
    }
  }
}