Probability Theory and Related Fields

, Volume 154, Issue 3, pp 661–675

Support convergence in the single ring theorem


DOI: 10.1007/s00440-011-0380-5

Cite this article as:
Guionnet, A. & Zeitouni, O. Probab. Theory Relat. Fields (2012) 154: 661. doi:10.1007/s00440-011-0380-5


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


Random matrices Non-commutative measure Schwinger–Dyson equation 

Mathematics Subject Classification (2010)

15A52 (46L50, 46L54) 

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.UMPA, CNRS UMR 5669, ENS LyonLyonFrance
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  3. 3.Faculty of MathematicsWeizmann InstituteRehovotIsrael

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