The asymptotic eigenvalue-distribution for a certain class of random matrices

Abstract For an n × n Hermitean matrix A with eigenvalues λ1, …, λn the eigenvalue-distribution is defined by G(x, A) := 1 n · number {λi: λi ⩽ x} for all real x. Let An for n = 1, 2, … be an n × n matrix, whose entries aik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, R , p) with the same distribution Fa. Suppose that all moments E | a | k, k = 1, 2, … are finite, E a=0 and E | a | 2. Let M(A)= ∑ σ=1 s θ σ P σ (A,A ∗ ) with complex numbers θσ and finite products Pσ of factors A and A ∗ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G0(x) = G1x) + G2(x), where G1 is a step function and G2 is absolutely continuous, such that with probability 1 G(x, M( A n n 1 2 )) converges to G0(x) as n → ∞ for all continuity points x of G0. The density g of G2 vanishes outside a finite interval. There are only finitely many jumps of G1. Both, G1 and G2, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices ( E. P. Wigner, Random matrices in physics, SIAM Review 9 (1967) , 1–23). The examples A r A ∗r , A r + A ∗r , A r A ∗r ± A ∗r A r , r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form lim sup n→∞ ∑ i=1 n |γ i (n) | 2 ⧹‖A n ‖ 2 ⩽ 0.8228… of Schur's inequality for the eigenvalues λi(n) of An holds. Consequently random matrices do not tend to be normal matrices for large n.