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Support convergence in the single ring theorem
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  • Published: 06 July 2011

Support convergence in the single ring theorem

  • Alice Guionnet1 &
  • Ofer Zeitouni2,3 

Probability Theory and Related Fields volume 154, pages 661–675 (2012)Cite this article

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  • 14 Citations

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Abstract

We study the eigenvalues of non-normal square matrices of the form A n  = U n T n V n with U n , V n independent Haar distributed on the unitary group and T n real diagonal. We show that when the empirical measure of the eigenvalues of T n converges, and T n satisfies some technical conditions, all these eigenvalues lie in a single ring.

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Author information

Authors and Affiliations

  1. UMPA, CNRS UMR 5669, ENS Lyon, 46 allée d’Italie, 69007, Lyon, France

    Alice Guionnet

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Ofer Zeitouni

  3. Faculty of Mathematics, Weizmann Institute, POB26, Rehovot, 76100, Israel

    Ofer Zeitouni

Authors
  1. Alice Guionnet
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  2. Ofer Zeitouni
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Corresponding author

Correspondence to Ofer Zeitouni.

Additional information

The work of this author was partially supported by NSF Grant DMS-0804133 and by a Grant from the Israel Science Foundation. This work was partially supported by the ANR Project ANR-08-BLAN-0311-01. Attribute ANR support to A. Guionnet, NSF and Israel Science foundation to O. Zeitouni.

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Cite this article

Guionnet, A., Zeitouni, O. Support convergence in the single ring theorem. Probab. Theory Relat. Fields 154, 661–675 (2012). https://doi.org/10.1007/s00440-011-0380-5

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  • Received: 09 January 2011

  • Revised: 07 June 2011

  • Published: 06 July 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0380-5

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Keywords

  • Random matrices
  • Non-commutative measure
  • Schwinger–Dyson equation

Mathematics Subject Classification (2010)

  • 15A52 (46L50, 46L54)
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