{
  "id": "1206.2251",
  "version": "v2",
  "published": "2012-06-11T15:18:58.000Z",
  "updated": "2012-06-26T21:49:57.000Z",
  "title": "A Necessary and Sufficient Condition for Edge Universality of Wigner matrices",
  "authors": [
    "Ji Oon Lee",
    "Jun Yin"
  ],
  "comment": "34 pages, 0 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we prove a necessary and sufficient condition for Tracy-Widom law of Wigner matrices. Consider $N \\times N$ symmetric Wigner matrices $H$ with $H_{ij} = N^{-1/2} x_{ij}$, whose upper right entries $x_{ij}$ $(1\\le i< j\\le N)$ are $i.i.d.$ random variables with distribution $\\mu$ and diagonal entries $x_{ii}$ $(1\\le i\\le N)$ are $i.i.d.$ random variables with distribution $\\wt \\mu$. The means of $\\mu$ and $\\wt \\mu$ are zero, the variance of $\\mu$ is 1, and the variance of $\\wt \\mu $ is finite. We prove that Tracy-Widom law holds if and only if $\\lim_{s\\to \\infty}s^4\\p(|x_{12}| \\ge s)=0$. The same criterion holds for Hermitian Wigner matrices.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-06-26T21:49:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sufficient condition",
      "edge universality",
      "random variables",
      "tracy-widom law holds",
      "hermitian wigner matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.2251O"
    }
  }
}