{
  "id": "2005.02254",
  "version": "v1",
  "published": "2020-05-05T14:50:45.000Z",
  "updated": "2020-05-05T14:50:45.000Z",
  "title": "Fluctuations of extreme eigenvalues of sparse Erdős-Rényi graphs",
  "authors": [
    "Yukun He",
    "Antti Knowles"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider a class of sparse random matrices which includes the adjacency matrix of the Erd\\H{o}s-R\\'enyi graph $\\mathcal{G}(N,p)$. We show that if $N^{\\varepsilon} \\leq Np \\leq N^{1/3-\\varepsilon}$ then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result [19] on the fluctuations of the extreme eigenvalues from $Np \\geq N^{2/9 + \\varepsilon}$ down to the optimal scale $Np \\geq N^{\\varepsilon}$. The main technical achievement of our proof is a rigidity bound of accuracy $N^{-1/2-\\varepsilon} \\, (Np)^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from [9,19,24]. Our result is the last missing piece, added to [8, 12, 19, 24], of a complete description of the eigenvalue fluctuations of sparse random matrices for $Np \\geq N^{\\varepsilon}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-05T14:50:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "05C50",
      "60B20",
      "15B52"
    ],
    "keywords": [
      "sparse erdős-rényi graphs",
      "extreme eigenvalues",
      "sparse random matrices",
      "nontrivial eigenvalues away",
      "complete description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}