{
  "id": "1306.2140",
  "version": "v1",
  "published": "2013-06-10T08:50:52.000Z",
  "updated": "2013-06-10T08:50:52.000Z",
  "title": "Heat Kernel Empirical Laws on $\\mathbb{U}_N$ and $\\mathbb{GL}_N$",
  "authors": [
    "Todd Kemp"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "This paper studies the empirical measures of eigenvalues and singular values for random matrices drawn from the heat kernel measures on the unitary groups $\\mathbb{U}_N$ and the general linear groups $\\mathbb{GL}_N$, for $N\\in\\mathbb{N}$. It establishes the strongest known convergence results for the empirical eigenvalues in the $\\mathbb{U}_N$ case, and the first known almost sure convergence results for the eigenvalues and singular values in the $\\mathbb{GL}_N$ case. The limit noncommutative distribution associated to the heat kernel measure on $\\mathbb{GL}_N$ is identified as the projection of a flow on an infinite-dimensional polynomial space. These results are then strengthened from variance estimates to $L^p$ estimates for even integers $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-10T08:50:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "46L54"
    ],
    "keywords": [
      "heat kernel empirical laws",
      "heat kernel measure",
      "singular values",
      "random matrices drawn",
      "general linear groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.2140K"
    }
  }
}