{
  "id": "2505.11993",
  "version": "v2",
  "published": "2025-05-17T13:11:41.000Z",
  "updated": "2025-06-23T08:29:30.000Z",
  "title": "Delocalization of random band matrices at the edge",
  "authors": [
    "Fan Yang",
    "Jun Yin"
  ],
  "comment": "60 pages. We corrected several typos and relocated some of the proofs to the appendix. arXiv admin note: text overlap with arXiv:2503.11382",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider $N\\times N$ Hermitian random band matrices $H=(H_{xy})$, whose entries are centered complex Gaussian random variables. The indices $x,y$ range over the $d$-dimensional discrete torus $(\\mathbb Z/L\\mathbb Z)^d$ with $d\\in \\{1,2\\}$ and $N=L^d$. The variance profile $S_{xy}=\\mathbb E|h_{xy}|^2$ exhibits a banded structure: specifically, $S_{xy}=0$ whenever the distance $|x-y|$ exceeds a band width parameter $W\\le L$. Let $W=L^\\alpha$ for some exponent $0<\\alpha\\le 1$. We show that as $\\alpha$ increases from $\\mathbf 1_{d=1}/2$ to $1-d/6$, the range of energies corresponding to delocalized eigenvectors gradually expands from the bulk toward the entire spectrum. More precisely, we prove that eigenvectors associated with energies $E$ satisfying $2 - |E| \\gg N^{-c_{d,\\alpha}}$ are delocalized, where the exponent $c_{d,\\alpha}$ is given by $c_{d,\\alpha} = 2\\alpha - 1$ in dimension 1 and $c_{d,\\alpha} = \\alpha$ in dimension 2. Furthermore, when $\\alpha > 1-d/6$, all eigenvectors of $H$ become delocalized. We further establish quantum unique ergodicity for delocalized eigenvectors, as well as a rigidity estimate for the eigenvalues. Our findings extend previous results -- established in the bulk regime for one-dimensional (1D) (arXiv:2501.01718) and two-dimensional (2D) (arXiv:2503.07606) random band matrices -- to the entire spectrum, including the spectral edges. They also complement the results of arXiv:0906.4047 and arXiv:2401.00492, which concern the edge eigenvalue statistics for 1D and 2D random band matrices.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-06-23T08:29:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "delocalization",
      "hermitian random band matrices",
      "entire spectrum",
      "centered complex gaussian random variables",
      "2d random band matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}