{
  "id": "1605.08767",
  "version": "v1",
  "published": "2016-05-27T19:35:40.000Z",
  "updated": "2016-05-27T19:35:40.000Z",
  "title": "Local law and Tracy-Widom limit for sparse random matrices",
  "authors": [
    "Ji Oon Lee",
    "Kevin Schnelli"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erd\\H{o}s-R\\'enyi graph model $G(N,p)$. We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erd\\H{o}s-R\\'{e}nyi graph this establishes the Tracy-Widom fluctuations of the second largest eigenvalue for $p\\gg N^{-1/3}$ with a deterministic shift of order $(Np)^{-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-27T19:35:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L54",
      "60B20"
    ],
    "keywords": [
      "sparse random matrices",
      "local law",
      "tracy-widom limit",
      "deterministic shift",
      "spectral edge"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}