{
  "id": "1604.05709",
  "version": "v1",
  "published": "2016-04-19T19:54:51.000Z",
  "updated": "2016-04-19T19:54:51.000Z",
  "title": "Universality of random matrices with correlated entries",
  "authors": [
    "Ziliang Che"
  ],
  "comment": "36 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider an $N$ by $N$ real symmetric random matrix $X=(x_{ij})$ where $\\mathbb{E}x_{ij}x_{kl}=\\xi_{ijkl}$. Under the assumption that $(\\xi_{ijkl})$ is the discretization of a piecewise Lipschitz function and that the correlation is short-ranged we prove that the empirical spectral measure of $X$ converges to a probability measure. The Stieltjes transform of the limiting measure can be obtained by solving a functional equation. Under the slightly stronger assumption that $(x_{ij})$ has a strictly positive definite covariance matrix, we prove a local law for the empirical measure down to the optimal scale $\\text{Im} z \\gtrsim N^{-1}$. The local law implies delocalization of eigenvectors. As another consequence we prove that the eigenvalue statistics in the bulk agrees with that of the GOE.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-19T19:54:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "correlated entries",
      "universality",
      "real symmetric random matrix",
      "local law implies delocalization",
      "strictly positive definite covariance matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160405709C"
    }
  }
}