We study the rate of convergence of the empirical spectral distribution of products of independent non-Hermitian random matrices to the power of the Circular Law. The distance to the deterministic limit distribution will be measured in terms of a uniform Kolmogorov-like distance. First, we prove that for products of Ginibre matrices, the optimal rate is given by , which is attained with overwhelming probability up to a logarithmic correction. Avoiding the edge, the rate of convergence of the mean empirical spectral distribution is even faster. Second, we show that also products of matrices with independent entries attain this optimal rate in the bulk up to a logarithmic factor. In the case of Ginibre matrices, we apply a saddlepoint approximation to a double contour integral representation of the density and in the case of matrices with independent entries we make use of techniques from local laws.
Financial support by the German Research Foundation (DFG) through the IRTG 2235 is gratefully acknowledged.
The author would like to thank Thorsten Neuschel and Friedrich Götze for valuable suggestions, Mario Kieburg for helpful discussions regarding the saddle-point analysis and the referees for very useful feedback and remarks.
"Rate of convergence for products of independent non-Hermitian random matrices." Electron. J. Probab. 26 1 - 24, 2021. https://doi.org/10.1214/21-EJP625