A rate of convergence for the circular law for the complex Ginibre ensemble

Abstract: We prove rates of convergence for the circular law for the complex Ginibre ensemble. Specifically, we bound the expected $L_p$-Wasserstein distance between the empirical spectral measure of the normalized complex Ginibre ensemble and the uniform measure on the unit disc, both in expectation and almost surely. For $1 \le p \le 2$, the bounds are of the order $n^{-1/4}$, up to logarithmic factors.