{
  "id": "1112.0752",
  "version": "v3",
  "published": "2011-12-04T13:55:51.000Z",
  "updated": "2014-01-13T13:23:05.000Z",
  "title": "Random matrices: Law of the determinant",
  "authors": [
    "Hoi H. Nguyen",
    "Van Vu"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/12-AOP791 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2014, Vol. 42, No. 1, 146-167",
  "doi": "10.1214/12-AOP791",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\\det A_n|$ satisfies a central limit theorem. More precisely, \\begin{eqnarray*}\\sup_{x\\in {\\mathbf {R}}}\\biggl|{\\mathbf {P}}\\biggl(\\frac{\\log(|\\det A_n|)-({1}/{2})\\log (n-1)!}{\\sqrt{({1}/{2})\\log n}}\\le x\\biggr)-{\\mathbf {P}}\\bigl(\\mathbf {N}(0,1)\\le x\\bigr)\\biggr|\\\\\\qquad\\le\\log^{-{1}/{3}+o(1)}n.\\end{eqnarray*}",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-01-13T13:23:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random matrices",
      "determinant",
      "independent real random variables",
      "central limit theorem",
      "random matrix"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.0752N"
    }
  }
}