PUB - Publikationen an der Universität Bielefeld

The aim of this thesis is to investigate the rate of convergence of empirical spectral distributions of non-Hermitian random matrices with independent entries and their products. The distance to the deterministic limiting distribution will be measured in terms of a uniform Kolmogorov-like metric. We will show that the optimal rate of convergence to the Circular Law is determined by Ginibre matrices and is given by n^{-1/2}. For products of Ginibre matrices, the optimal rate of convergence to powers of the Circular Law is shown to be n^{-1/2} as well. Interestingly, the rate of convergence of the mean empirical spectral distribution is even faster in the bulk of the spectrum. Furthermore, we develop an approach to study the rate of convergence for matrices with independent entries, which are not necessarily Gaussian. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law. Moreover, we show that also products of matrices with independent entries attain the optimal rate in the bulk up to a logarithmic factor. The robustness of this approach enables us to similarly obtain the same rate of convergence in terms of the classical two-dimensional Kolmogorov distance as well as for the empirical measure of the roots of random Weyl polynomials. Finally, we shall relate our result to the spectral radius of non-Hermitian random matrices and investigate its rate of convergence.