The Strong Elliptic Law. Twenty years later. Part I

Abstract

The structure of this survey is the following: at first we repeat in Part I the first 20 years old proof of the strong Elliptic law for random matrices Then in Part II we prove the strong Elliptic law for random matrices Ξ n of the general form, i.e. when their diagonal entries have nonzero expectations, and when we do not require the existence of the probability densities of the entries of random matrices, but instead of Lyapunov condition we require the boundedness of the absolute moments of the entries of the order 4 + δ , δ > 0. In this case the Elliptic law means that the support of the accompanying spectral density of eigenvalues looks like the picture of several galaxies made by telescope. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies. We also present the sand clock law density for the sum of diagonal complex matrix and the symmetric random matrix belonging the domain of attraction of the Semicircular law. After this we prove in the Part III the weak Elliptic law, when all entries can have nonzero expectations. At the end of survey we give the proof of the Elliptic law for the Unitary random matrices from the so called Class N 12 of random Unitary matrices and give the over turned stool law density for the sum of diagonal complex matrix A n and where Ξ n belongs to the domain of attraction of the Circular law. We also establish in Part III for Elliptic Law at the first time the convergency rate and the Central Limit Theorem. These statements are based on the VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Random Independent Arrays. 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 n Ξ n B n + C n ) are eigenvalues of the matrix A n Ξ n B n + C n , 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. We emphasize that the spectral theory of Hermitian random matrices is rather profound. For example, in 1975 in [3] V. Girko proved the general stochastic canonical equation for ACE (Asymptotically Constant Entries)-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 when n → ∞, 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 K 3 [3, 23] at the points x of continuity, 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 K 3 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 K 3 can be found by the method of successive approximations. For the first time in 1980[4] and in 1990 in [13, p.282] this equation was rewritten in the following form (here we use the simplest equation, when α = 0) J 0 ( x ) is the Bessel function which is equal to In [13] a technical improvement and a new proof of the uniqueness of solution of canonical equation K 3 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[4]. English translation: Ukrainian Math. J. 32 (1980), no. 6, 546–548 (1980). We remind readers the formulation of the STRONG ELLIPTIC LAW established twenty years ago: let be eigenvalues of random complex matrix whose pairs of entries are independent for every n and are given on the same probability space where ρ is a complex number, for some and the real and imaginary parts of random entries have the densities satisfying the Elliptic condition : for some β > 1 and there exist the densities of the random entries or the densities of the random entries satisfying the condition: for some β 1 > 1 is the normalized spectral function and χ is the indicator function. Then with probability one