{
  "id": "1012.1107",
  "version": "v1",
  "published": "2010-12-06T09:53:30.000Z",
  "updated": "2010-12-06T09:53:30.000Z",
  "title": "How many eigenvalues of a Gaussian random matrix are positive?",
  "authors": [
    "Satya N. Majumdar",
    "Céline Nadal",
    "Antonello Scardicchio",
    "Pierpaolo Vivo"
  ],
  "comment": "25 pages, 6 figures",
  "journal": "Phys. Rev. E 83, 041105 (2011)",
  "doi": "10.1103/PhysRevE.83.041105",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the probability distribution of the index ${\\mathcal N}_+$, i.e., the number of positive eigenvalues of an $N\\times N$ Gaussian random matrix. We show analytically that, for large $N$ and large $\\mathcal{N}_+$ with the fraction $0\\le c=\\mathcal{N}_+/N\\le 1$ of positive eigenvalues fixed, the index distribution $\\mathcal{P}({\\mathcal N}_+=cN,N)\\sim\\exp[-\\beta N^2 \\Phi(c)]$ where $\\beta$ is the Dyson index characterizing the Gaussian ensemble. The associated large deviation rate function $\\Phi(c)$ is computed explicitly for all $0\\leq c \\leq 1$. It is independent of $\\beta$ and displays a quadratic form modulated by a logarithmic singularity around $c=1/2$. As a consequence, the distribution of the index has a Gaussian form near the peak, but with a variance $\\Delta(N)$ of index fluctuations growing as $\\Delta(N)\\sim \\log N/\\beta\\pi^2$ for large $N$. For $\\beta=2$, this result is independently confirmed against an exact finite $N$ formula, yielding $\\Delta(N)= \\log N/2\\pi^2 +C+\\mathcal{O}(N^{-1})$ for large $N$, where the constant $C$ has the nontrivial value $C=(\\gamma+1+3\\log 2)/2\\pi^2\\simeq 0.185248...$ and $\\gamma=0.5772...$ is the Euler constant. We also determine for large $N$ the probability that the interval $[\\zeta_1,\\zeta_2]$ is free of eigenvalues. Part of these results have been announced in a recent letter [\\textit{Phys. Rev. Lett.} {\\bf 103}, 220603 (2009)].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-06T09:53:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "02.50.-r",
      "02.10.Yn",
      "24.60.-k"
    ],
    "keywords": [
      "gaussian random matrix",
      "associated large deviation rate function",
      "distribution",
      "positive eigenvalues",
      "index fluctuations"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review E",
      "year": 2011,
      "month": "Apr",
      "volume": 83,
      "number": 4,
      "pages": "041105"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011PhRvE..83d1105M"
    }
  }
}