{
  "id": "1009.0135",
  "version": "v3",
  "published": "2010-09-01T09:37:58.000Z",
  "updated": "2011-06-19T06:24:50.000Z",
  "title": "Large deviations of the extreme eigenvalues of random deformations of matrices",
  "authors": [
    "Florent Benaych-Georges",
    "Alice Guionnet",
    "Mylène Maïda"
  ],
  "comment": "44 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale $n$, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of $X_n$ converge to the edges of the support of the limiting measure and when we allow some eigenvalues of $X_n$, that we call outliers, to converge out of the bulk. We can also generalise our results to the case when $X_n$ is random, with law proportional to $e^{- n Trace V(X)}\\ud X,$ for $V$ growing fast enough at infinity and any perturbation of finite rank.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-06-19T06:24:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extreme eigenvalues",
      "random deformations",
      "real diagonal deterministic matrix",
      "random finite rank matrix",
      "large deviation principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.0135B"
    }
  }
}