{
  "id": "1307.5753",
  "version": "v2",
  "published": "2013-07-22T16:04:43.000Z",
  "updated": "2014-10-21T14:00:36.000Z",
  "title": "On Eigenvectors of Random Band Matrices with Large Band",
  "authors": [
    "Stefan Steinerberger"
  ],
  "comment": "This paper has been withdrawn by the author due to a gap in the proof",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study random, symmetric $N \\times N$ band matrices with a band of size $W$ and Bernoulli random variables as entries. This interpolates between nearest neighbour interaction $W = 1$ and Wigner matrices $W = N$. Eigenvectors are known to be localized for $W \\ll N^{1/8}$, delocalized for $W \\gg N^{4/5}$ and it is conjectured that the transition for the bulk occurs at $W \\sim N^{1/2}$. Eigenvalues in the spectral edge change their behavior at $W \\sim N^{5/6}$ but nothing is known about the associated eigenvectors. We show that up to $W \\ll N^{5/7}$ any random matrix has with large probability some eigenvectors in the spectral edge, which either exhibit mass concentration or interact strongly on a small scale.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-22T16:04:43.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-21T14:00:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random band matrices",
      "large band",
      "eigenvectors",
      "bernoulli random variables",
      "spectral edge change"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.5753S"
    }
  }
}