{
  "id": "1312.0212",
  "version": "v2",
  "published": "2013-12-01T12:59:18.000Z",
  "updated": "2015-06-19T20:09:40.000Z",
  "title": "Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble",
  "authors": [
    "Y. V. Fyodorov",
    "B. A. Khoruzhenko",
    "N. J. Simm"
  ],
  "comment": "47 pages, 3 figures. This version includes a completely rewritten introduction and statement of results. Improved exposition in Section 5 and minor changes to the Appendix",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "The goal of this paper is to establish a relation between characteristic polynomials of $N \\times N$ GUE random matrices $\\mathcal{H}$ as $N \\to \\infty$, and Gaussian processes with logarithmic correlations. First, we introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of $D_{N}(z):=-\\log|\\det(zI-\\mathcal{H})|$ on mesoscopic scales as $N \\to \\infty$. By employing a Fourier integral representation, we show how this implies a continuous analogue of a result by Diaconis and Shahshahani \\cite{DS94}. On the macroscopic scale, $D_{N}(x)$ gives rise to yet another type of Gaussian process with logarithmic correlations. We give an explicit construction of the latter in terms of a Chebyshev-Fourier random series.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-01T12:59:18.000Z",
      "comment": "55 pages, 4 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-19T20:09:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "60F05"
    ],
    "keywords": [
      "fractional brownian motion",
      "gaussian unitary ensemble",
      "gaussian process",
      "logarithmic correlations",
      "logarithmic increment structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.0212F"
    }
  }
}