{
  "id": "1904.07140",
  "version": "v1",
  "published": "2019-04-15T15:50:18.000Z",
  "updated": "2019-04-15T15:50:18.000Z",
  "title": "Bulk eigenvalue fluctuations of sparse random matrices",
  "authors": [
    "Yukun He"
  ],
  "comment": "30 pages, comments are welcome!",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider a class of sparse random matrices, which includes the adjacency matrix of Erd\\H{o}s-R\\'enyi graphs $\\mathcal G(N,p)$ for $p \\in [N^{\\varepsilon-1},N^{-\\varepsilon}]$. We identify the joint limiting distributions of the eigenvalues away from 0 and the spectral edges. Our result indicates that unlike Wigner matrices, the eigenvalues of sparse matrices satisfy central limit theorems with normalization $N\\sqrt{p}$. In addition, the eigenvalues fluctuate simultaneously: the correlation of two eigenvalues of the same/different sign is asymptotically 1/-1. We also prove a CLT for mesoscopic linear statistics of sparse matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-15T15:50:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "15B52",
      "60B20",
      "05C50"
    ],
    "keywords": [
      "sparse random matrices",
      "bulk eigenvalue fluctuations",
      "matrices satisfy central limit theorems",
      "sparse matrices satisfy central limit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}