Random Block Matrices and Matrix Orthogonal Polynomials

Holger Dette () and Bettina Reuther ()
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Holger Dette: Ruhr-Universität Bochum
Bettina Reuther: Ruhr-Universität Bochum

Journal of Theoretical Probability, 2010, vol. 23, issue 2, 378-400

Abstract: Abstract In this paper we consider random block matrices, which generalize the general beta ensembles recently investigated by Dumitriu and Edelmann (J. Math. Phys. 43:5830–5847, 2002; Ann. Inst. Poincaré Probab. Stat. 41:1083–1099, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence, we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature.

Keywords: Random block matrices; Matrix orthogonal polynomials; Asymptotic eigenvalue distribution; Strong uniform approximation; Chebyshev matrix polynomials; 60F15; 15A15 (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s10959-008-0189-z

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