{
  "id": "1503.03533",
  "version": "v1",
  "published": "2015-03-11T23:18:52.000Z",
  "updated": "2015-03-11T23:18:52.000Z",
  "title": "Mesoscopic linear statistics of Wigner matrices",
  "authors": [
    "A. Lodhia",
    "N. J. Simm"
  ],
  "comment": "32 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study linear spectral statistics of $N \\times N$ Wigner random matrices $\\mathcal{H}$ on mesoscopic scales. Under mild assumptions on the matrix entries of $\\mathcal{H}$, we prove that after centering and normalizing, the trace of the resolvent $\\mathrm{Tr}(\\mathcal{H}-z)^{-1}$ converges to a stationary Gaussian process as $N \\to \\infty$ on scales $N^{-1/3} \\ll \\mathrm{Im}(z) \\ll 1$ and explicitly compute the covariance structure. The limit process is related to certain regularizations of fractional Brownian motion and logarithmically correlated fields appearing in \\cite{FKS13}. Finally, we extend our results to general mesoscopic linear statistics and prove that the limiting covariance is given by the $H^{1/2}$-norm of the test functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-11T23:18:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20"
    ],
    "keywords": [
      "wigner matrices",
      "general mesoscopic linear statistics",
      "study linear spectral statistics",
      "wigner random matrices",
      "stationary gaussian process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}