Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution

Let X be a real $(β=1)$ or complex $(β=2)$ Ginibre ensemble. Let \σᵢ\1≤ i≤ n be the eigenvalues of $X,$ and Zₙ be some rescaled version of maxᵢ σᵢ. It was proved that Zₙ converges weakly to the Gumbel distribution Λ_β with distribution function e^-β2e⁻ˣ. We further prove that x∈ R|P(Zₙ ≤ x)-e^-β2e⁻ˣ|=25log log n/4e log n(1+o(1)) and W₁(L(Zₙ), Λ_β)=25log log n/4log n(1+o(1)) for sufficiently large n, where L(Zₙ) is the distribution of Zₙ and W₁ is the Wasserstein distance. Similar results hold for maxᵢ |σᵢ|. Furthermore, the convergence rates of the complex Ginibre ensemble are universal for complex iid random matrices under certain moment conditions on entries.