{
  "id": "1412.3085",
  "version": "v1",
  "published": "2014-12-09T20:14:05.000Z",
  "updated": "2014-12-09T20:14:05.000Z",
  "title": "Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices",
  "authors": [
    "Olivier Marchal"
  ],
  "comment": "46 pages, 10 figures",
  "categories": [
    "math-ph",
    "math.DS",
    "math.MP",
    "math.PR"
  ],
  "abstract": "The purpose of this article is to study the eigenvalues $u_1^{\\, t}=e^{it\\theta_1},\\dots,u_N^{\\,t}=e^{it\\theta_N}$ of $U^t$ where $U$ is a large $N\\times N$ random unitary matrix and $t>0$. In particular we are interested in the typical times $t$ for which all the eigenvalues are simultaneously close to $1$ in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first orders of the large $N$ asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge towards an exponential distribution when $N$ is large. Numeric simulations are provided along the paper to illustrate the results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-09T20:14:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "60B20",
      "15B05"
    ],
    "keywords": [
      "random unitary matrices",
      "recurrence times",
      "matrix models",
      "random matrix integral",
      "loop equations techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}