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The norm of polynomials in large random and deterministic matrices
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  • Published: 21 June 2011

The norm of polynomials in large random and deterministic matrices

  • Camille Male1 

Probability Theory and Related Fields volume 154, pages 477–532 (2012)Cite this article

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Abstract

Let \({{\bf X}_N =(X_1^{(N)}, \ldots, X_p^{(N)})}\) be a family of N × N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices \({{\bf Y}_N =(Y_1^{(N)}, \ldots, Y_q^{(N)})}\) , possibly random but independent of X N , for which the operator norm of \({P({\bf X}_N, {\bf Y}_N, {\bf Y}_N^*)}\) converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y N and of the polynomials P, we get for a large class of matrices the “no eigenvalues outside a neighborhood of the limiting spectrum” phenomena. We give examples of diagonal matrices Y N for which the convergence holds. Convergence of the operator norm is shown to hold for block matrices, even with rectangular Gaussian blocks, a situation including non-white Wishart matrices and some matrices encountered in MIMO systems.

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Authors and Affiliations

  1. Ecole Normale Supérieure de Lyon, Unité de Mathématiques pures et appliquées, UMR 5669, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    Camille Male

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  1. Camille Male
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Correspondence to Camille Male.

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With an appendix by Dimitri Shlyakhtenko.

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Male, C. The norm of polynomials in large random and deterministic matrices. Probab. Theory Relat. Fields 154, 477–532 (2012). https://doi.org/10.1007/s00440-011-0375-2

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  • Received: 22 June 2010

  • Revised: 25 May 2011

  • Published: 21 June 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0375-2

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Keywords

  • Random matrix
  • Free probability
  • Strong asymptotic
  • Freeness
  • C*-algebra

Mathematics Subject Classification (2000)

  • 15A52
  • 46L54
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