Rate of convergence to the Circular Law via smoothing inequalities for log-potentials
F. Götze, J. Jalowy, Random Matrices: Theory and Applications 10 (2020).
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Journal Article
| Published
| English
Author
Götze, Friedrich;
Jalowy, Jonas
Abstract
<jats:p> The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by [Formula: see text]. 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 in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials. </jats:p>
Publishing Year
Journal Title
Random Matrices: Theory and Applications
Volume
10
Issue
03
Article Number
2150026
LibreCat-ID
Cite this
Götze F, Jalowy J. Rate of convergence to the Circular Law via smoothing inequalities for log-potentials. Random Matrices: Theory and Applications. 2020;10(03). doi:10.1142/s201032632150026x
Götze, F., & Jalowy, J. (2020). Rate of convergence to the Circular Law via smoothing inequalities for log-potentials. Random Matrices: Theory and Applications, 10(03), Article 2150026. https://doi.org/10.1142/s201032632150026x
@article{Götze_Jalowy_2020, title={Rate of convergence to the Circular Law via smoothing inequalities for log-potentials}, volume={10}, DOI={10.1142/s201032632150026x}, number={032150026}, journal={Random Matrices: Theory and Applications}, publisher={World Scientific Pub Co Pte Lt}, author={Götze, Friedrich and Jalowy, Jonas}, year={2020} }
Götze, Friedrich, and Jonas Jalowy. “Rate of Convergence to the Circular Law via Smoothing Inequalities for Log-Potentials.” Random Matrices: Theory and Applications 10, no. 03 (2020). https://doi.org/10.1142/s201032632150026x.
F. Götze and J. Jalowy, “Rate of convergence to the Circular Law via smoothing inequalities for log-potentials,” Random Matrices: Theory and Applications, vol. 10, no. 03, Art. no. 2150026, 2020, doi: 10.1142/s201032632150026x.
Götze, Friedrich, and Jonas Jalowy. “Rate of Convergence to the Circular Law via Smoothing Inequalities for Log-Potentials.” Random Matrices: Theory and Applications, vol. 10, no. 03, 2150026, World Scientific Pub Co Pte Lt, 2020, doi:10.1142/s201032632150026x.