Convergence rate of extreme eigenvalues of Ginibre ensembles to the Gumbel distribution

Let [Formula: see text] be a real ([Formula: see text]) or complex ([Formula: see text]) Ginibre ensemble. Let [Formula: see text] be the eigenvalues of [Formula: see text], and let [Formula: see text] be a suitably rescaled version of [Formula: see text]. It has been proved that [Formula: see text] converges weakly to the Gumbel distribution [Formula: see text] with distribution function [Formula: see text]. We further prove that for sufficiently large [Formula: see text], [Formula: see text] and for the complex Ginibre ensemble [Formula: see text] where [Formula: see text] denotes the distribution of [Formula: see text] and [Formula: see text] is the Wasserstein distance. Similar results hold for the spectral radius [Formula: see text] for both the real and complex Ginibre ensembles. Furthermore, the convergence rates for the complex Ginibre ensemble are universal for complex i.i.d. random matrices under certain moment conditions on the entries.