{
  "id": "2501.01718",
  "version": "v1",
  "published": "2025-01-03T09:23:18.000Z",
  "updated": "2025-01-03T09:23:18.000Z",
  "title": "Delocalization of One-Dimensional Random Band Matrices",
  "authors": [
    "Horng-Tzer Yau",
    "Jun Yin"
  ],
  "comment": "81 pages, 14 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider an $ N \\times N$ Hermitian one-dimensional random band matrix with band width $W > N^{1 / 2 + \\varepsilon} $ for any $ \\varepsilon > 0$. In the bulk of the spectrum and in the large $ N $ limit, we obtain the following results: (i) The semicircle law holds up to the scale $ N^{-1 + \\varepsilon} $ for any $ \\varepsilon > 0 $. (ii) All $ L^2 $- normalized eigenvectors are delocalized, meaning their $ L^\\infty$ norms are simultaneously bounded by $ N^{-\\frac{1}{2} + \\varepsilon} $ with overwhelming probability, for any $ \\varepsilon > 0 $. (iii) Quantum unique ergodicity holds in the sense that the local $ L^2 $ mass of eigenvectors becomes equidistributed with high probability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-03T09:23:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "82B44"
    ],
    "keywords": [
      "hermitian one-dimensional random band matrix",
      "delocalization",
      "quantum unique ergodicity holds",
      "semicircle law holds",
      "high probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 81,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}