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On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.
1 The eigenvalue distribution function
"... For an N × N matrix AN, the eigenvalue distribution function 1 (e.d.f.) F AN (x) is defined as ..."
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For an N × N matrix AN, the eigenvalue distribution function 1 (e.d.f.) F AN (x) is defined as
RANDOM MATRICES AND ORTHOGONAL POLYNOMIALS
, 2006
"... The central question of the theory of random matrices is to determine the asymptotic behavior of the eigenvalues of large random symmetric or Hermitian matrices. In the case of the unitary Gaussian ensemble, i.e. the space of Hermitian matrices equipped with a unitarily invariant Gaussian probabilit ..."
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The central question of the theory of random matrices is to determine the asymptotic behavior of the eigenvalues of large random symmetric or Hermitian matrices. In the case of the unitary Gaussian ensemble, i.e. the space of Hermitian matrices equipped with a unitarily invariant Gaussian probability, Mehta’s formulae express the eigenvalue density in terms of the ChristoffelDarboux kernel of the Hermite polynomials. In fact orthogonal polynomials are a powerful tool in this theory. We will present in this course methods in the theory of random matrices which