Rate of Convergence to the Circular Law
A talk in the AG Mathematische Physik series by
Jonas Jalowy from Bielefeld University
| Abstract: | It is well known that the (complex) empirical spectral distribution of a
non-Hermitian random matrix with i.i.d. entries will converge to the
uniform distribution on the complex disc as the size of the matrix tends
to infinity. In this talk, we investigate the rate of convergence to the
Circular Law in terms of a uniform, 2-dimensional Kolmogorov-like
distance. The optimal rate of convergence is determined by the Ginibre
ensemble and is given by $n^{-1/2}$. I will present a smoothing
inequality for complex measures that quantitatively relates the
Kolmogorov-like distance to the concentration of logarithmic potentials.
Combining it with results from local circular laws, it is applied to
prove nearly optimal rate of convergence to the circular law with
overwhelming probability. Furthermore I will relate the result to other
distances, present an analogue for the empirical root measure of Weyl
random polynomials with independent coefficients and discuss a possible
generalization for products of independent matrices. The talk is based
on joint work with Friedrich Götze. |