Limiting spectral distribution of sum of unitary and orthogonal matrices

Anirban Basak (Stanford University)
Amir Dembo (Stanford University)


We show that the empirical eigenvalue measure for sum of $d$ independent Haar distributed $n$-dimensional unitary matrices, converge for $n \rightarrow \infty$ to the Brown measure of the free sum of $d$ Haar unitary operators. The same applies for independent Haar distributed $n$-dimensional orthogonal matrices. As a byproduct of our approach, we relax the requirement of uniformly bounded imaginary part of Stieltjes transform of $T_n$ that is made in [Guionnet, Krishnapur, Zeitouni; Theorem 1].

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Pages: 1-19

Publication Date: August 10, 2013

DOI: 10.1214/ECP.v18-2466


  • Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer. An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010. xiv+492 pp. ISBN: 978-0-521-19452-5 MR2760897
  • Biane, Philippe; Lehner, Franz. Computation of some examples of Brown's spectral measure in free probability. Colloq. Math. 90 (2001), no. 2, 181--211. MR1876844
  • Bordenave, Charles; Chafaï, Djalil. Around the circular law. Probab. Surv. 9 (2012), 1--89. MR2908617
  • Brown, L. G. Lidskiĭ's theorem in the type ${\rm II}$ case. Geometric methods in operator algebras (Kyoto, 1983), 1--35, Pitman Res. Notes Math. Ser., 123, Longman Sci. Tech., Harlow, 1986. MR0866489
  • Forrester, P. J. Log-gases and random matrices. London Mathematical Society Monographs Series, 34. Princeton University Press, Princeton, NJ, 2010. xiv+791 pp. ISBN: 978-0-691-12829-0 MR2641363
  • Girko, V. L. The circular law. (Russian) Teor. Veroyatnost. i Primenen. 29 (1984), no. 4, 669--679. MR0773436
  • Guionnet, Alice; Krishnapur, Manjunath; Zeitouni, Ofer. The single ring theorem. Ann. of Math. (2) 174 (2011), no. 2, 1189--1217. MR2831116
  • Guionnet, Alice; Zeitouni, Ofer. Support convergence in the single ring theorem. Probab. Theory Related Fields 154 (2012), no. 3-4, 661--675. MR3000558
  • Haagerup, Uffe; Larsen, Flemming. Brown's spectral distribution measure for $R$-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176 (2000), no. 2, 331--367. MR1784419
  • Hiai, Fumio; Petz, Dénes. Asymptotic freeness almost everywhere for random matrices. Acta Sci. Math. (Szeged) 66 (2000), no. 3-4, 809--834. MR1804226
  • C. Male. The distribution of traffics and their free product: an asymptotic freeness theorem for random matrices and a central limit theorem. Preprint, arXiv:1111.4662v3.
  • Meckes, Elizabeth S.; Meckes, Mark W. Concentration and convergence rates for spectral measures of random matrices. Probab. Theory Related Fields 156 (2013), no. 1-2, 145--164. MR3055255
  • Mehta, Madan Lal. Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. xviii+688 pp. ISBN: 0-12-088409-7 MR2129906
  • Nica, Alexandru; Speicher, Roland. $R$-diagonal pairs—a common approach to Haar unitaries and circular elements. Free probability theory (Waterloo, ON, 1995), 149--188, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997. MR1426839
  • M. Rudelson, and R. Vershynin. Invertibility of random matrices: unitary and orthogonal transformation. Journal of the AMS, to appear.
  • Śniady, Piotr. Random regularization of Brown spectral measure. J. Funct. Anal. 193 (2002), no. 2, 291--313. MR1929504
  • Tao, Terence; Vu, Van. Random matrices: universality of ESDs and the circular law. With an appendix by Manjunath Krishnapur. Ann. Probab. 38 (2010), no. 5, 2023--2065. MR2722794

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