{
  "id": "1412.2445",
  "version": "v1",
  "published": "2014-12-08T04:50:35.000Z",
  "updated": "2014-12-08T04:50:35.000Z",
  "title": "Fluctuations of Linear Eigenvalue Statistics of Random Band Matrices",
  "authors": [
    "Indrajit Jana",
    "Koushik Saha",
    "Alexander Soshnikov"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study the fluctuation of linear eigenvalue statistics of Random Band Matrices defined by $M_{n}=\\frac{1}{\\sqrt{b_{n}}}W_{n}$, where $W_{n}$ is a $n\\times n$ band Hermitian random matrix of bandwidth $b_{n}$, i.e., the diagonal elements and only first $b_{n}$ off diagonal elements are nonzero. Also variances of the matrix elmements are upto a order of constant. We study the linear eigenvalue statistics $\\mathcal{N}(\\phi)=\\sum_{i=1}^{n}\\phi(\\lambda_{i})$ of such matrices, where $\\lambda_{i}$ are the eigenvalues of $M_{n}$ and $\\phi$ is a sufficiently smooth function. We prove that $\\sqrt{\\frac{b_{n}}{n}}[\\mathcal{N}(\\phi)-\\mathbb{E} \\mathcal{N}(\\phi)]\\stackrel{d}{\\to} N(0,V(\\phi))$ for $b_{n}>>\\sqrt{n}$, where $V(\\phi)$ is given in the Theorem 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-08T04:50:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear eigenvalue statistics",
      "fluctuation",
      "band hermitian random matrix",
      "diagonal elements",
      "smooth function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.2445J"
    }
  }
}