{
  "id": "1508.03111",
  "version": "v1",
  "published": "2015-08-13T03:42:31.000Z",
  "updated": "2015-08-13T03:42:31.000Z",
  "title": "Empirical Distributions of Eigenvalues of Product Ensembles",
  "authors": [
    "Tiefeng Jiang",
    "Yongcheng Qi"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the limits of the empirical distributions of the eigenvalues of two $n$ by $n$ matrices as $n$ goes to infinity. The first one is the product of $m$ i.i.d. (complex) Ginibre ensembles, and the second one is that of truncations of $m$ independent Haar unitary matrices with sizes $n_j\\times n_j$ for $1\\leq j \\leq m$. Assuming $m$ depends on $n$, by using the special structures of the eigenvalues of the two matrices we developed, explicit limits of spectral distributions are obtained regardless of the speed of $m$ compared to $n$. In particular, we show a rich feature of the limits for the second matrix as $n_j/n$'s vary. Some general results on arbitrary rotation-invariant determinantal point processes are also derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-13T03:42:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "empirical distributions",
      "product ensembles",
      "eigenvalues",
      "arbitrary rotation-invariant determinantal point processes",
      "independent haar unitary matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150803111J"
    }
  }
}