{
  "id": "2102.00963",
  "version": "v1",
  "published": "2021-02-01T16:44:58.000Z",
  "updated": "2021-02-01T16:44:58.000Z",
  "title": "Spectrum of Random $d$-regular Graphs Up to the Edge",
  "authors": [
    "Jiaoyang Huang",
    "Horng-Tzer Yau"
  ],
  "comment": "67 pages, 4 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\\geq3$. We prove that, with probability $1-N^{-1+{\\mathfrak c}}$ for any ${\\mathfrak c} >0$, the following two properties hold as $N \\to \\infty$ provided that $d\\geq3$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten-McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in $N$, i.e. $\\lambda_2, |\\lambda_N|\\leq 2+N^{-\\Omega(1)}$. (ii) All eigenvectors of random $d$-regular graphs are completely delocalized.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-01T16:44:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "05C80"
    ],
    "keywords": [
      "regular graphs",
      "polynomial error bound",
      "classical eigenvalue locations",
      "properties hold",
      "kesten-mckay distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 67,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}