{
  "id": "2009.13143",
  "version": "v1",
  "published": "2020-09-28T08:44:16.000Z",
  "updated": "2020-09-28T08:44:16.000Z",
  "title": "Eigenvector distribution in the critical regime of BBP transition",
  "authors": [
    "Zhigang Bao",
    "Dong Wang"
  ],
  "comment": "52 pages, 19 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "In this paper, we study the random matrix model of Gaussian Unitary Ensemble (GUE) with fixed-rank (aka spiked) external source. We will focus on the critical regime of the Baik-Ben Arous-P\\'ech\\'e (BBP) phase transition and establish the distribution of the eigenvectors associated with the leading eigenvalues. The distribution is given in terms of a determinantal point process with extended Airy kernel. Our result can be regarded as an eigenvector counterpart of the BBP eigenvalue phase transition (arXiv:math/0403022). The derivation of the distribution makes use of the recently re-discovered eigenvector-eigenvalue identity, together with the determinantal point process representation of the GUE minor process with external source.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-28T08:44:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15A18"
    ],
    "keywords": [
      "critical regime",
      "bbp transition",
      "eigenvector distribution",
      "determinantal point process representation",
      "external source"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}