A Simple Approach to Global Regime of the Random Matrix Theory

We study the normalized trace $g_n(z)=n^{-1} \mbox{tr} \, (H-zI)^{-1}$ of the resolvent of $n\times n$ real symmetric matrices $H=\big[(1+\delta_{jk})W_{jk}/\sqrt n\big]_{j,k=1}^n$ assuming that their entries are independent but not necessarily identically distributed random variables. We develop a rigorous method of asymptotic analysis of moments of $g_n(z)$ for $|\Im z| \ge \eta_0$ where $\eta_0$ is determined by the second moment of $W_{jk}$. By using this method we find t...

For a large class of symmetric random matrices with correlated entries, selected from stationary random fields of centered and square integrable variables, we show that the limiting distribution of eigenvalue counting measure always exists and we describe it via an equation satisfied by its Stieltjes transform. No rate of convergence to zero of correlations is imposed, therefore the process is allowed to have long memory. In particular, if the symmetrized matrices are constru...

Several applications of the moment method in random matrix theory, especially, to local eigenvalue statistics at the spectral edges, are surveyed, with emphasis on a modification of the method involving orthogonal polynomials.

This is an elementary review, aimed at non-specialists, of results that have been obtained for the limiting distribution of eigenvalues and for the operator norms of real symmetric random matrices via the method of moments. This method goes back to a remarkable argument of Eugen Wigner some sixty years ago which works best for independent matrix entries, as far as symmetry permits, that are all centered and have the same variance. We then discuss variations of this classical ...

This is a brief survey of classical and recent results about the typical behavior of eigenvalues of large random matrices, written for mathematicians and others who study and use matrices but may not be accustomed to thinking about randomness.

We describe the resolvent approach for the rigorous study of the mescoscopic regime of Hermitian matrix spectra. We present results reflecting the universal behavior of the smoothed density of eigenvalue distribution of large random matrices

This paper studies the behaviour of the empirical eigenvalue distribution of large random matrices W_N W_N* where W_N is a ML x N matrix, whose M block lines of dimensions L x N are mutually independent Hankel matrices constructed from complex Gaussian correlated stationary random sequences. In the asymptotic regime where M \rightarrow \infty, N \rightarrow +\infty and ML/N \rightarrow c > 0, it is shown using the Stieltjes transform approach that the empirical eigenvalue dis...

We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. To be more precise, we consider random Hermitian matrices with independent Gaussian entries $M_{ij}, i\leq j$ with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large $N$ limit, various limiting distributions depending on both the eigenvalues of the mat...

We investigate the universality of singular value and eigenvalue distributions of matrix valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution and prove universality under general conditions for singular value and eigenvalue distributions of products of independent matrices from spherical ensembles.

In this paper we show that the empirical eigenvalue distribution of any sample covariance matrix generated by independent copies of a stationary regular sequence has a limiting distribution depending only on the spectral density of the sequence. We characterize this limit in terms of Stieltjes transform via a certain simple equation. No rate of convergence to zero of the covariances is imposed. If the entries of the stationary sequence are functions of independent random vari...