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.

中文翻译：

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通过平滑对数势的不等式收敛到循环定律的速率

本文的目的是研究循环定律的 Kolmogorov 距离与具有独立条目的非厄米随机矩阵的经验谱分布。最佳收敛速度由 Ginibre 集成确定，由下式给出n-1/2. 显示了将均匀 Kolmogorov 样距离与对数电位浓度定量关联的复杂测量的平滑不等式。将其与局部循环定律的结果相结合，我们将其应用于证明在 Kolmogorov 距离上接近循环定律的最佳收敛速度。此外，我们表明，对于 Weyl 随机多项式的根的经验测度，同样的收敛速度是成立的。