{
  "id": "2401.00492",
  "version": "v1",
  "published": "2023-12-31T13:25:37.000Z",
  "updated": "2023-12-31T13:25:37.000Z",
  "title": "Edge statistics for random band matrices",
  "authors": [
    "Dang-Zheng Liu",
    "Guangyi Zou"
  ],
  "comment": "61 pages, 8 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider Hermitian and symmetric random band matrices $H=(\\sigma_{xy}A_{xy})$ on the $d$-dimensional lattice $\\left(\\mathbb{Z}/{L\\mathbb{Z}}\\right)^d$, where $A_{xy}=\\overline{A_{yx}}$ are independent uniformly distributed random variables on $S^1$ or $\\{+1, -1\\}$, and the variance profile $\\sigma^{2}_{xy}$ is characterized by the bandwidth $W$ and $\\alpha$-stable density with $\\alpha\\in (0,2]$. We investigate local eigenvalue statistics at the spectral edge as $W\\to \\infty$ and observe the critical dimension $d_c=3\\alpha$ and the critical bandwidth $W_c=L^{(1-\\frac{d}{3\\alpha})_{+}}$, possibly with a $\\log L$ correction when $d=\\alpha$ or $2\\alpha$. In the Hermitian case, we establish that (i) when $d<2\\alpha$, GUE edge, interpolating, and Poisson statistics emerge in the supercritical ($W\\gg W_c$), critical ($W\\sim W_c$), and subcritical ($W\\ll W_c$) regimes, respectively; (ii) when $d\\ge 2\\alpha$, as long as $W\\ge L^{\\frac{1}{3}+\\epsilon}$ for a small constant $\\epsilon>0$, GUE edge universality holds. In the symmetric case, we also establish similar but subtle phenomena. In both $d=1$ and $\\alpha=2$, the subcritical and supercritical results have been proven by Sodin for the band model with a cutoff variance profile \\cite{sodin2010spectral}. Our proof builds upon Sodin's program and new techniques of taming the singularity of Feynman diagrams and graph integrals through a connection to the $\\phi^3$ model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-31T13:25:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20"
    ],
    "keywords": [
      "edge statistics",
      "gue edge universality holds",
      "symmetric random band matrices",
      "independent uniformly distributed random variables",
      "local eigenvalue statistics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}