{
  "id": "1601.02188",
  "version": "v1",
  "published": "2016-01-10T08:35:36.000Z",
  "updated": "2016-01-10T08:35:36.000Z",
  "title": "Limit Laws for Random Matrices from Traffic-Free Probability",
  "authors": [
    "Benson Au"
  ],
  "comment": "35 pages, 14 figures",
  "categories": [
    "math.PR",
    "math.CO",
    "math.OA"
  ],
  "abstract": "We consider a random matrix ensemble that interpolates between the Wigner matrices and the random Markov matrices studied by Bryc, Dembo, and Jiang. For a real Wigner matrix $\\mathbf{W}_n$, let $\\mathbf{D}_n$ be the diagonal matrix whose entries correspond to the row sums of $\\mathbf{W}_n$, i.e., $\\mathbf{D}_n(i,i) = \\sum_{j=1}^n \\mathbf{W}_n(i,j)$. For $p,q\\in\\mathbb{R}$, we study the asymptotic behavior of the matrices $\\mathbf{M}_{n,p,q}=p\\mathbf{W}_n+q\\mathbf{D}_n$ as $n\\to\\infty$. By realizing $\\mathbf{M}_{n,p,q}$ as a traffic of $\\mathbf{W}_n$ in the sense of Male, we show that when the entries of $\\mathbf{W}_n$ have finite moments of all orders, independent $\\mathbf{M}_{n,p,q}$ are asymptotically traffic-free with stable universal limiting traffic distribution. We obtain the limiting spectral distributions of the ensemble $(\\mathbf{M}_{n,p,q})_{p,q\\in\\mathbb{R}}$ as a consequence via the traffic-free central limit theorem. The continuity of the limiting spectral distribution in the parameters $p,q$ allows us to incorporate additional randomness: if $(X_n)$ and $(Y_n)$ are sequences of real-valued random variables converging a.s. to $X$ and $Y$ respectively, we show that the empirical spectral distributions $\\mu(\\mathbf{M}_{n,X_n,Y_n})$ converge weakly a.s. to the random free convolution $\\mathcal{SC}(0,X^2)\\boxplus\\mathcal{N}(0,Y^2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-10T08:35:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L53",
      "46L54",
      "60B10",
      "60B20",
      "60E07",
      "60F05"
    ],
    "keywords": [
      "random matrix",
      "traffic-free probability",
      "limit laws",
      "universal limiting traffic distribution",
      "limiting spectral distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160102188A"
    }
  }
}