{
  "id": "1603.08845",
  "version": "v1",
  "published": "2016-03-29T17:00:23.000Z",
  "updated": "2016-03-29T17:00:23.000Z",
  "title": "Delocalization at small energy for heavy-tailed random matrices",
  "authors": [
    "Charles Bordenave",
    "Alice Guionnet"
  ],
  "comment": "47 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove that the eigenvectors associated to small enough eigenvalues of an heavy-tailed symmetric random matrix are delocalized with probability tending to one as the size of the matrix grows to infinity. The delocalization is measured thanks to a simple criterion related to the inverse participation ratio which computes an average ratio of L4 and L2-norms of vectors. In contrast, as a consequence of a previous result, for random matrices with sufficiently heavy tails, the eigenvectors associated to large enough eigenvalues are localized according to the same criterion. The proof is based on a new analysis of the fixed point equation satisfied asymptotically by the law of a diagonal entry of the resolvent of this matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-29T17:00:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heavy-tailed random matrices",
      "small energy",
      "delocalization",
      "point equation",
      "inverse participation ratio"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160308845B"
    }
  }
}