{
  "id": "1405.2581",
  "version": "v2",
  "published": "2014-05-11T20:56:19.000Z",
  "updated": "2016-12-06T07:28:10.000Z",
  "title": "Random Matrices with Log-Range Correlations, and Log-Sobolev Inequalities",
  "authors": [
    "Todd Kemp",
    "David Zimmermann"
  ],
  "categories": [
    "math.FA",
    "math.PR"
  ],
  "abstract": "Let $X_N$ be a symmetric $N\\times N$ random matrix whose $\\sqrt{N}$-scaled centered entries are uniformly square integrable. We prove that if the entries of $X_N$ can be partitioned into independent subsets each of size $o(\\log N)$, then the empirical eigenvalue distribution of $X_N$ converges weakly to its mean in probability. This significantly extends the best previously known results on convergence of eigenvalues for matrices with correlated entries (where the partition subsets are blocks and of size $O(1)$.) we prove this result be developing a new log-Sobolev inequality, generalizing the first author's introduction of mollified log-Sobolev inequalities: we show that if $\\mathbf{Y}$ is a bounded random vector and $\\mathbf{Z}$ is a standard normal random vector independent from $\\mathbf{Y}$, then the law of $\\mathbf{Y}+t\\mathbf{Z}$ satisfies a log-Sobolev inequality for all $t>0$, and we give bounds on the optimal log-Sobolev constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-11T20:56:19.000Z",
      "title": "Bounds for logarithmic Sobolev constants for Gaussian convolutions of compactly supported measures",
      "abstract": "We give upper bounds for optimal constants in logarithmic Sobolev inequalities for convolutions of compactly supported measures on $\\mathbb{R}^n$ with a Gaussian measure. We investigate tightness of these bounds by examining some examples. We then revisit and improve (by significantly weakening hypotheses) the statement and proof in \\cite{Zi13} that, under certain hypotheses, the empirical law of eigenvalues of a sequence of symmetric random matrices converges to its mean in probability. In particular, the universality theorem holds even when there is some mild long range dependence between the entries.",
      "comment": null,
      "journal": null,
      "doi": null,
      "authors": [
        "David Zimmermann"
      ]
    },
    {
      "version": "v2",
      "updated": "2016-12-06T07:28:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compactly supported measures",
      "logarithmic sobolev constants",
      "gaussian convolutions",
      "symmetric random matrices converges",
      "mild long range dependence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.2581Z"
    }
  }
}