The Strong Circular Law. Twenty years later. Part I

The structure of this survey is the following: first we repeat the first, now 20 years old, proof of the Strong Circular Law for non Hermitian random matrices under the assumptions that the probability densities of the random entries exist and they satisfy the Lyapunov condition. (see [4] and the sketch of the proof of this law in the paper V-transform, Dopovidi Akademii nauk Ukrainskoi RSR, Seria A: Fizyko-Matematychni ta technichni nauky, 1982, N3, pp.5-6.(see[2])). Then we prove the Strong V-Law for random matrices of general form, i.e. the entries of matrices have nonzero expectations and different variances. In this case the V-Law means that the support of the accompanying spectral density of eigenvalues looks like several water or mercury drops on the table. According to the distances between the centers of these drops, these drops might not touch each other or can merge making fanciful shapes represented at the end of the paper, Figures 13–16. If the distances between the centers of these drops are large enough, we have several separate almost circular drops. In Part II we prove the weak Circular law, (where we do not require the above assumptions on the densities and Lyapunov condition) but instead of these two conditions we require that the Weak Circular Law condition is fulfilled: for some In Part III of the survey we give the formulation of the Circular Law for Unitary random matrices from the so called Class N₁₂ of random Unitary matrices(see[20]). In this case the Circular law means that the support of the accompanying spectral density of eigenvalues looks like several drops of water, oil or mercury on the table and inside of some drops it is possible that some dry circles appear. If the distances between the centers of these drops are large enough we have several almost circular drops as for corresponding description of limit spectral density for the Hermitian random matrices. In Part II we prove the following Weak Global Circular Law( Weak V -Law) which generalizes the Strong Global Circular Law (Strong V -Law): For every n, let the random entries of the complex matrix be independent, the Weak Circular Law condition is fulfilled and

where and are square complex nonrandom matrices, det A_n ≠ 0, det B_n ≠ 0. Moreover we require the so-called Central Limit V -condition and Border V -condition are fulfilled. See the precise formulation of these condition in the Section 1 of this paper and in the Part II. Roughly speaking, the Central Limit V -condition means that we can apply the central limit theorem for the heights of random parallelogram (obtained from the random matrix). The Border V -condition means that we can exclude from V -transform some small region between two regions were we apply two different method of limit theorems.

Then, in probability, for almost all x and y

where

λ_k are eigenvalues of the matrix A_nΞ_nB_n + C_n; the Global probability density p_n,α (t, s) = (∂²/∂t∂s)F_n,α (t, s) is equal to

where the analytic function m_n (y; t; s) satisfies the canonical equation K₂₆(see[12, 13, 20])

In particular, we have gathered in this paper three proofs of the STRONG CIRCULAR LAW: 1. the first proof based on the V -regularization of the determinant of random matrix and on the application of an inequality of the Berry-Esseen type[4]. 2. the second proof based on the central limit theorem for random determinants and on the replacement a square matrix by rectangular random matrix[11]. 3. the third proof based on the regularization of random determinant and replacement square matrix by rectangular matrix[18]. Therefore, we have proved triply the Circular law:

Let be a complex random matrix whose entries are defined on a common probability space, are independent for any n = 1, 2, ... and are such that and . Further, assume that there exist either densities of the real parts of the random entries (or densities of the imaginary parts of the random entries and, for some δ > 0) and β > 1

Then, with probability one,

where

is a normalized spectral function and λ_k are eigenvalues of the matrix