Abstract
Beginning from this chapter we consider the simple case of analytic functions of random matrices \( {A_n} + \;{U_n}{B_n}U_n^* \). The problem of describing limit normalized spectral functions of matrices \( {A_n} + \;{U_n}{B_n}U_n^* \), where A n and B n are Hermitian nonrandom matrices and U n is a unitary or orthogonal random matrix distributed by probability Haar measure, has a long history. Such problem was announced by L. A. Pastur in 1973 in [Pas1]. Later on the problem was attacked by several authors [Lar], [NS1,2], [VDN]. At last Pastur and Vasilchuk [PaV] found the solution of this problem and the system of equations for the Stieltjes transform of n.s.f. of matrices \( {A_n} + \;{U_n}{B_n}U_n^* \). In this chapter we investigate a similar problem for several classes of random unitary matrices using the REFORM (REsolvents FORmulas and Martingale) method.
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