{
  "id": "0910.0775",
  "version": "v1",
  "published": "2009-10-05T14:23:01.000Z",
  "updated": "2009-10-05T14:23:01.000Z",
  "title": "The Index Distribution of Gaussian Random Matrices",
  "authors": [
    "Satya N. Majumdar",
    "Celine Nadal",
    "Antonello Scardicchio",
    "Pierpaolo Vivo"
  ],
  "comment": "4 pages Revtex, 4 .eps figures included",
  "journal": "Phys. Rev. Lett. 103, 220603 (2009)",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.dis-nn",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N_{+}) of a random NxN matrix belonging to Gaussian orthogonal (\\beta=1), unitary (\\beta=2) or symplectic (\\beta=4) ensembles. The distribution of the fraction of positive eigenvalues c=N_{+}/N scales, for large N, as Prob(c,N)\\simeq\\exp[-\\beta N^2 \\Phi(c)] where the rate function \\Phi(c), symmetric around c=1/2 and universal (independent of $\\beta$), is calculated exactly. The distribution has non-Gaussian tails, but even near its peak at c=1/2 it is not strictly Gaussian due to an unusual logarithmic singularity in the rate function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-10-05T14:23:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.40.-a",
      "02.10.Yn",
      "05.45.Mt",
      "24.60.-k"
    ],
    "keywords": [
      "gaussian random matrices",
      "index distribution",
      "rate function",
      "positive eigenvalues",
      "unusual logarithmic singularity"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review Letters",
      "doi": "10.1103/PhysRevLett.103.220603",
      "year": 2009,
      "month": "Nov",
      "volume": 103,
      "number": 22,
      "pages": 220603
    },
    "note": {
      "typesetting": "RevTeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009PhRvL.103v0603M"
    }
  }
}