{
  "id": "1807.01562",
  "version": "v1",
  "published": "2018-07-04T13:20:20.000Z",
  "updated": "2018-07-04T13:20:20.000Z",
  "title": "Random band matrices in the delocalized phase, II: Generalized resolvent estimates",
  "authors": [
    "Paul Bourgade",
    "Fan Yang",
    "Horng-Tzer Yau",
    "Jun Yin"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of $N\\times N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances $\\mathbb E |H_{ij}|^2$ form a band matrix with typical band width $1\\ll W\\ll N$. We consider the generalized resolvent of $H$ defined as $G(Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix such that $Z_{ij}=\\left(z 1_{1\\leq i \\leq W}+\\widetilde z 1_{ i > W} \\right) \\delta_{ij}$, with two distinct spectral parameters $z\\in \\mathbb C_+:=\\{z\\in \\mathbb C:{\\rm Im} z>0\\}$ and $\\widetilde z\\in \\mathbb C_+\\cup \\mathbb R$. In this paper, we prove a sharp bound for the local law of the generalized resolvent $G$ for $W\\gg N^{3/4}$. This bound is a key input for the proof of delocalization and bulk universality of random band matrices in \\cite{PartI}. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in \\cite{PartIII}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-04T13:20:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "82B44"
    ],
    "keywords": [
      "random band matrices",
      "band matrix",
      "generalized resolvent estimates",
      "delocalized phase",
      "part series abut delocalization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}