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+B_nU_n(ε)C_n, where A_n,B_n and C_n are nonrandom matrices, and U_n(ε) is random matrix from class C₁₂[26] or from the following class C₁₃ of distributions of random matrices: where are independent random real matrices whose entries are independent for every n, and for a certain δ > 0 the Lyapunov condition is fulfilled: This problem has been considered in some publications for matrices A_n + U_n, where Unitary matrix U_n 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 U_n and a nonrandom matrix A_n has not been discovered. Many conclusions of various kinds have been presented in the literature (see [13–27]) for random matrices A_n + Ξ_nB_n, where Ξ_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.

We give the formulation of the Circular Law for random matrices A_n + B_nU_n(ε)C_n, where A_n,B_n and C_n are nonrandom matrices, and U_n(ε) is from the so called Class C₁₂ of random Unitary matrices (see[26]). In this case the Circular law means that the support of accompanying spectral density p(x, y) of eigenvalues ofA_n + B_nU_n(ε)C_n looks like several drops of mercury on a table, and inside of some drops it is possible that some dry circles appear. We call this support of limit density p(x, y) the Mercury support. If the distances between the centers of these drops are large enough we have several separate almost circular drops as for corresponding description of limit spectral density for the Hermitian random matrices [28, 29]. According to the distances between the centers of these drops, these drops might not touch each other or can merge creating fanciful shapes represented at the end of the paper, Figures 1–16. The analogy with the Circular Law is the following: for a simple Unitary matrix U_n from class C₁ all its eigenvalues are distributed asymptotically uniformly on the circumference and, evidently, this limit distribution does not coincide with the uniform distribution on the circle. But if we will add to Unitary matrix U_n any diagonal nonrandom matrix A_n = (δ_ija_j) then if the distances between diagonal entries a_j are large enough we have almost the same picture of limit distribution of eigenvalues of U_n + A_n if Unitary matrix U_n would be equal to the random matrix Ξ_n with random independent entries satisfying the Global Circular Law (see Part I of this paper). The rough explanation of this phenomenon is the following: the expected scalar products of the vector row or vector column of matrix Ξ_n have the order n⁻¹. Therefore, the matrix Ξ_n looks like orthogonal random matrix. In this paper we use for matrices A_n+B_nU_n(ε)C_n the triply regularized V -transform[2, 5],(VICTORIA-transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Randomly Independent Arrays)

where α > 0, ε ≠= 0, γ ≠= 0 are regularization parameters , τ = t + is is a complex number and µ_n {x,G_n(t, s, γ, ε)} is the normalized spectral function of G-matrix:

Such a V -transformation was used for the first time in 1982–1985 in [2,3,5]. Canonical equations for the limit distribution function of normalized spectral functions of some matrices A_n + B_nU_n and the following V-domainG_δ, δ > 0(see [27, pp.142, 173]) of the pseudodistribution of eigenvalues of matrices A_n + B_nU_n are found:

where are eigenvalues of the matrix.