{
  "id": "2212.11634",
  "version": "v1",
  "published": "2022-12-22T12:03:42.000Z",
  "updated": "2022-12-22T12:03:42.000Z",
  "title": "Extreme eigenvalues of Log-concave Ensemble",
  "authors": [
    "Zhigang Bao",
    "Xiaocong Xu"
  ],
  "categories": [
    "math.PR",
    "math.FA",
    "math.MG",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "In this paper, we consider the log-concave ensemble of random matrices, a class of covariance-type matrices $XX^*$ with isotropic log-concave $X$-columns. A main example is the covariance estimator of the uniform measure on isotropic convex body. Non-asymptotic estimates and first order asymptotic limits for the extreme eigenvalues have been obtained in the literature. In this paper, with the recent advancements on log-concave measures \\cite{chen, KL22}, we take a step further to locate the eigenvalues with a nearly optimal precision, namely, the spectral rigidity of this ensemble is derived. Based on the spectral rigidity and an additional ``unconditional\" assumption, we further derive the Tracy-Widom law for the extreme eigenvalues of $XX^*$, and the Gaussian law for the extreme eigenvalues in case strong spikes are present.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-22T12:03:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extreme eigenvalues",
      "log-concave ensemble",
      "first order asymptotic limits",
      "spectral rigidity",
      "isotropic convex body"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}