Rate of convergence to the Circular Law via smoothing inequalities for log-potentials

Götze F, Jalowy J (2021)
Random Matrices: Theory and Applications 10(3): 2150026.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
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 n-1/2. 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.
Stichworte
Non-Hermitian random matrices; log-determinant; logarithmic potential; circular law; rate of convergence; smoothing inequality
Erscheinungsjahr
2021
Zeitschriftentitel
Random Matrices: Theory and Applications
Band
10
Ausgabe
3
Art.-Nr.
2150026
ISSN
2010-3263
eISSN
2010-3271
Page URI
https://pub.uni-bielefeld.de/record/2957345

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Götze F, Jalowy J. Rate of convergence to the Circular Law via smoothing inequalities for log-potentials. Random Matrices: Theory and Applications. 2021;10(3): 2150026.
Götze, F., & Jalowy, J. (2021). Rate of convergence to the Circular Law via smoothing inequalities for log-potentials. Random Matrices: Theory and Applications, 10(3), 2150026. https://doi.org/10.1142/S201032632150026X
Götze, F., and Jalowy, J. (2021). Rate of convergence to the Circular Law via smoothing inequalities for log-potentials. Random Matrices: Theory and Applications 10:2150026.
Götze, F., & Jalowy, J., 2021. Rate of convergence to the Circular Law via smoothing inequalities for log-potentials. Random Matrices: Theory and Applications, 10(3): 2150026.
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, 2021, : 2150026.
Götze, F., Jalowy, J.: Rate of convergence to the Circular Law via smoothing inequalities for log-potentials. Random Matrices: Theory and Applications. 10, : 2150026 (2021).
Götze, Friedrich, and Jalowy, Jonas. “Rate of convergence to the Circular Law via smoothing inequalities for log-potentials”. Random Matrices: Theory and Applications 10.3 (2021): 2150026.

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