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.

arXiv commentFinal version: to appear in Annales de la Faculté des Sciences de Toulouse. Previous comments: New result added giving almost sure convergence rates in addition to rates in expectation. One figure added to illustrate an initial segment with respect to the spiral order. Several references to related work added