{
  "id": "2012.13218",
  "version": "v1",
  "published": "2020-12-24T12:32:06.000Z",
  "updated": "2020-12-24T12:32:06.000Z",
  "title": "Functional Central Limit Theorems for Wigner Matrices",
  "authors": [
    "László Erdős",
    "Giorgio Cipolloni",
    "Dominik Schröder"
  ],
  "comment": "46 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the fluctuations of regular functions $f$ of a Wigner matrix $W$ viewed as an entire matrix $f(W)$. Going beyond the well studied tracial mode, $\\mathrm{Tr}[f(W)]$, which is equivalent to the the customary linear statistics of eigenvalues, we show that $\\mathrm{Tr}[f(W)]$ is asymptotically normal for any bounded deterministic matrix $A$. We identify three different and asymptotically independent modes of this fluctuation, corresponding to the tracial part, the traceless diagonal part and the off-diagonal part of $f(W)$ in the entire mesoscopic regime, where we find that the off-diagonal modes fluctuate on a much smaller scale than the tracial mode. In addition, we determine the fluctuations in the Eigenstate Thermalisation Hypothesis [Deutsch 1991], i.e. prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral averaging. In particular, in the macroscopic regime our result generalises [Lytova 2013] to complex $W$ and to all crossover ensembles in between. The main technical inputs are the recent multi-resolvent local laws with traceless deterministic matrices from the companion paper [Erd\\H{o}s, Cipolloni, Schr\\\"oder 2020].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-24T12:32:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52"
    ],
    "keywords": [
      "functional central limit theorems",
      "wigner matrix",
      "deterministic matrix",
      "tracial mode",
      "customary linear statistics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}