The Circular Law. Twenty years later. Part III

In this paper we apply the REFORM method[ 3, 4] to the deduction of the system of canonical equations for normalized spectral functions of the matrices A(n)+ BnUn(epsilon) C-n, where A(n), B-n and Cn are nonrandom matrices, and U-n(epsilon) is random matrix from class C-12[ 26] or from the following class C-13 of distributions of random matrices: U-n(epsilon) = lim beta down arrow 0 (SIC)(n()(SIC)(n)* (SIC)(n) + I-n beta)1/2 + epsilon H-n, where epsilon not equal 0, (SIC)(n) = (xi((n))(n)(ij))(i, j= 1) and H-n = (eta((n))(ij))(n)(i, j= 1) are independent random real matrices whose entries. (xi((n))(ij)) i, j = 1 and. (eta((n))(ij)) i, j = 1 are independent for every n, E. (n) ij = 0, E (xi((n))(ij)) ij = 0, E vertical bar xi((n))(2)(i, j broken vertical bar) = n(-) E eta((n)) (ij) = 0, E|n(ij)((n))|(2) = n(-1), i, j = 1,..., n, and for a certain delta > 0 the Lyapunov condition is fulfilled: E vertical bar(eta((n))(ij) root n vertical bar(2+delta) < c < infinity, i, j = 1,..., n. E xi((n))(ij) =0 E vertical bar(eta((n))(ij) root n vertical bar(2+delta) < c < infinity, i, j = 1,..., n. This problem has been considered in some publications for matrices A(n) + U-n, where Unitary matrix Un from the class C1[ 26] is distributed by Haar measure, at the "ad hoc" level, without a strong proof, on the basis of heuristic calculations. Therefore, the behavior of limit n. s. f. of the sum of a random Unitary matrix Un and a nonrandom matrix An has not been discovered. Many conclusions of various kinds have been presented in the literature (see [13-27]) for random matrices A(n) + (SIC)(n)B-n, where.n is the random matrix with independent entries, concerning, for example, the effectiveness of the REFORM method and the role of the martingale difference representation for the resolvent of random matrices.