Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution

Consider the complex Ginibre ensemble, whose eigenvalues are $({\lambda }_i{)}_{1\le i\le n}$ and the spectral radius $R_n={\max }_{1\le i\le n}\mathrm{\mid }{\lambda }_i\mathrm{\mid }\mathrm{.}$ Set $X_n=\sqrt{4{\gamma }_n}(R_n-\sqrt{n}-\frac{1}{2}\sqrt{{\gamma }_n})$ and $F_n$ be its distribution function, where ${\gamma }_n=\log n-2\log (\sqrt{2\pi }\log n)\mathrm{.}$ It was proved in \cite{Rider 2003} that $F_n$ converges weakly to the Gumbel distribution $\mathrm{\Lambda }\mathrm{.}$ We prove in further in this paper that $$\underset{n\to \mathrm{\infty }}{\lim }\frac{\log n}{\log \log n}{W}_1(F_n,\mathrm{\Lambda })=2$$ and the Berry-Esseen bound $$\underset{n\to \mathrm{\infty }}{\lim }\frac{\log n}{\log \log n}\underset{x\in \mathbb{R}}{\sup }\mathrm{\mid }{F}_n(x)-e^{-e^{-x}}\mathrm{\mid }=\frac{2}{e}\mathrm{.}$$