Electronic Communications in Probability

Spectral measures of powers of random matrices

Elizabeth Meckes and Mark Meckes

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This paper considers the empirical spectral measure of a power of a random matrix drawn uniformly from one of the compact classical matrix groups. We give sharp bounds on the $L_p$-Wasserstein distances between this empirical measure and the uniform measure on the circle, which show a smooth transition in behavior when the power increases and yield rates on almost sure convergence when the dimension grows. Along the way, we prove the sharp logarithmic Sobolev inequality on the unitary group.

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 78, 13 pp.

Accepted: 23 September 2013
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60E15: Inequalities; stochastic orderings 60F05: Central limit and other weak theorems

Uniform random matrices spectral measure Wasserstein distance logarithmic Sobolev inequality

This work is licensed under a Creative Commons Attribution 3.0 License.


Meckes, Elizabeth; Meckes, Mark. Spectral measures of powers of random matrices. Electron. Commun. Probab. 18 (2013), paper no. 78, 13 pp. doi:10.1214/ECP.v18-2551. https://projecteuclid.org/euclid.ecp/1465315617

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