Alexander Tikhomirov

THE CIRCULAR LAW FOR RANDOM MATRICES

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending results to a larger class of sparse matrices.

Limit Theorems for Spectra of Random Matrices with Martingale Structure

We study classical ensembles of real symmetric random matrices introduced by Eugene Wigner. We discuss Stein’s method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure of high dimensional matrices. The method is based on a differential equation for the density of the semicircle law.

The rate of convergence for spectra of GUE and LUE matrix ensembles

We obtain optimal bounds of order O(n−1) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively.

Asymptotic Distribution of Quadratic Forms and Applications

AbstractWe consider the quadratic formsQ

Asymptotic Spectra of Matrix-Valued Functions of Independent Random Matrices and Free Probability

\sum\limits_{\mathop {1 \leqslant j,k \leqslant N}\limits_{j \ne k} } {a_{jk} }

On the rate of convergence of the expected spectral distribution function of a Wigner matrix to the semi-circular law

XjXk+

On minimal singular values of random matrices with correlated entries

\sum\limits_{j = 1}^N {a_{jj} }