The Strong Circular Law. Twenty years later. Part II

We follow the main strategy of the theory of limit theorems of the probability theory, i.e. we try to solve the problem of description of all limits of normalized spectral functions

where λ_k(A_nΞ_nB_n+C_n) are eigenvalues of the matrix A_n Ξ_nB_n+C_n, where A_n,B_n, and C_n are nonrandom matrices, under general (as only possible) conditions on the entries of random matrices Ξ_n, χ is the indicator function. In 1975 in [2] V. Girko proved the general stochastic canonical equation for ACE-symmetric matrices: Assume that for any n, the random entries of a symmetric matrix are independent and they are asymptotically constant entries (ACE), i.e., for any ε > 0,

and τ > 0 is an arbitrary constant, and that, for every 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1,

where the symbol ⟾ denotes the weak convergence of distribution functions when n → ∞,

and K (u, v, z) is a nondecreasing function with bounded variation in z and continuous in u and v in the domain 0 ≤ u, v ≤ 1. Then, with probability one, for almost all x,

where λ_k (Ξ_{n × n}) are eigenvalues, F(x) is a distribution function whose Stieltjes transform satisfies the relation

G_α (x, y, t), as a function of x, is a distribution function satisfying the regularized stochastic canonical equation K₃[20] at the points x of continuity,

ξ_α {G_α (∗, ∗, t) , z} is a random real functional whose Laplace transform of one-dimensional distribution is equal to

The integrand is defined at x = 0 by continuity as –sy. There exists a unique solution of the canonical equation K₃ in the class L of functions G_α (x, y, t) that are distribution functions of x (0 ≤ x ≤ 1) for any fixed 0 ≤ y ≤ 1, –∞ < t < ∞, such that, for any integer k > 0 and z, the function is analytic in t (excluding, possibly, the origin). The solution of the canonical equation K₃ can be found by the method of successive approximations.

For the first time in 1980 in [2] and in 1990 in [10, p.282] this equation was rewritten in the following form (here we use the simplest equation, when α = 0)

where is the Bessel function which is equal to

In [2, 10] a technical improvement and a new proof of the uniqueness of solution of canonical equation K₃ are presented, where m(s, t, z) has a unique representation in the family of integrable functions. The analytic details of the statement and of the proof are elaborate. English translation: Ukrainian Math. J. 32 (1980), no. 6, 546–548 (1981).

In the second part of survey we give the proof of the Global Circular Law (which is known also as Weak V -Law) without conditions of the existence of probability densities of the entries of random matrices Ξ_n. We require only independence of random entries of random matrices, the equality of their variances and Weak V -condition like a Lyapunov condition. The proof of V -Law in this case becomes more complicated and instead of convergency with probability one (strong V -Law) we have convergence in probability (weak V -Law) for normalized spectral functions

where λ_k(A_n Ξ_nB_n+C_n) are eigenvalues of matrix A_n Ξ_nB_n+C_n, where A_n, B_n and C_n are nonrandom matrices. For the readers convenience we repeat in the first section of the second part of the paper the formulation of the Global Circular Law(Weak V -Law).