{
  "id": "1210.2904",
  "version": "v1",
  "published": "2012-10-10T13:13:17.000Z",
  "updated": "2012-10-10T13:13:17.000Z",
  "title": "Distribution of Schmidt-like eigenvalues for Gaussian Ensembles of the Random Matrix Theory",
  "authors": [
    "M. P. Pato",
    "G. Oshanin"
  ],
  "comment": "10 pages, 1 figure, submitted to JSTAT",
  "categories": [
    "cond-mat.dis-nn",
    "cond-mat.stat-mech"
  ],
  "abstract": "We analyze the form of the probability distribution function P_{n}^{(\\beta)}(w) of the Schmidt-like random variable w = x_1^2/(\\sum_{j=1}^n x^{2}_j/n), where x_j are the eigenvalues of a given n \\times n \\beta-Gaussian random matrix, \\beta being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual eigenvalue deviates from the arithmetic mean value of all eigenvalues of a given random matrix, and its distribution is calculated with respect to the ensemble of such \\beta-Gaussian random matrices. We show that in the asymptotic limit n \\to \\infty and for arbitrary \\beta the distribution P_{n}^{(\\beta)}(w) converges to the Mar\\v{c}enko-Pastur form, i.e., is defined as P_{n}^{(\\beta)}(w) \\sim \\sqrt{(4 - w)/w} for w \\in [0,4] and equals zero outside of the support. Furthermore, for Gaussian unitary (\\beta = 2) ensembles we present exact explicit expressions for P_{n}^{(\\beta=2)}(w) which are valid for arbitrary n and analyze their behavior.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-10T13:13:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random matrix theory",
      "gaussian ensembles",
      "schmidt-like eigenvalues",
      "individual eigenvalue deviates",
      "dyson symmetry index"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/1751-8113/46/11/115002",
      "journal": "Journal of Physics A Mathematical General",
      "year": 2013,
      "month": "Mar",
      "volume": 46,
      "number": 11,
      "pages": 115002
    },
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013JPhA...46k5002P"
    }
  }
}