Electronic Communications in Probability

Limiting spectral distribution of sum of unitary and orthogonal matrices

Anirban Basak and Amir Dembo

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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].

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 69, 19 pp.

Accepted: 10 August 2013
First available in Project Euclid: 7 June 2016

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Zentralblatt MATH identifier

Primary: 46L53: Noncommutative probability and statistics
Secondary: 60B10: Convergence of probability measures 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Random matrices limiting spectral distribution Haar measure Brown measure free convolution Stieltjes transform Schwinger-Dyson equation

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Basak, Anirban; Dembo, Amir. Limiting spectral distribution of sum of unitary and orthogonal matrices. Electron. Commun. Probab. 18 (2013), paper no. 69, 19 pp. doi:10.1214/ECP.v18-2466. https://projecteuclid.org/euclid.ecp/1465315608

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