{
  "id": "1205.5669",
  "version": "v3",
  "published": "2012-05-25T11:49:50.000Z",
  "updated": "2013-06-27T19:31:03.000Z",
  "title": "Delocalization and Diffusion Profile for Random Band Matrices",
  "authors": [
    "Laszlo Erdos",
    "Antti Knowles",
    "Horng-Tzer Yau",
    "Jun Yin"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider Hermitian and symmetric random band matrices $H = (h_{xy})$ in $d \\geq 1$ dimensions. The matrix entries $h_{xy}$, indexed by $x,y \\in (\\bZ/L\\bZ)^d$, are independent, centred random variables with variances $s_{xy} = \\E |h_{xy}|^2$. We assume that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In one dimension we prove that the eigenvectors of $H$ are delocalized if $W\\gg L^{4/5}$. We also show that the magnitude of the matrix entries $\\abs{G_{xy}}^2$ of the resolvent $G=G(z)=(H-z)^{-1}$ is self-averaging and we compute $\\E \\abs{G_{xy}}^2$. We show that, as $L\\to\\infty$ and $W\\gg L^{4/5}$, the behaviour of $\\E |G_{xy}|^2$ is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-06-27T19:31:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "82B44",
      "82C44"
    ],
    "keywords": [
      "diffusion profile",
      "symmetric random band matrices",
      "matrix entries",
      "delocalization",
      "higher dimensions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00220-013-1773-3",
      "journal": "Communications in Mathematical Physics",
      "year": 2013,
      "month": "Oct",
      "volume": 323,
      "number": 1,
      "pages": 367
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013CMaPh.323..367E"
    }
  }
}