{
  "id": "2305.01428",
  "version": "v1",
  "published": "2023-05-02T13:58:10.000Z",
  "updated": "2023-05-02T13:58:10.000Z",
  "title": "Edge Universality of Random Regular Graphs of Growing Degrees",
  "authors": [
    "Jiaoyang Huang",
    "Horng-Tzer Yau"
  ],
  "comment": "53 pages, 3 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We consider the statistics of extreme eigenvalues of random $d$-regular graphs, with $N^{\\mathfrak c}\\leq d\\leq N^{1/3-{\\mathfrak c}}$ for arbitrarily small ${\\mathfrak c}>0$. We prove that in this regime, the fluctuations of extreme eigenvalues are given by the Tracy-Widom distribution. As a consequence, about 69% of $d$-regular graphs have all nontrivial eigenvalues bounded in absolute value by $2\\sqrt{d-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-02T13:58:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "05C80"
    ],
    "keywords": [
      "random regular graphs",
      "edge universality",
      "growing degrees",
      "extreme eigenvalues",
      "tracy-widom distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}