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

Abstract

Consider the complex Ginibre ensemble, whose eigenvalues are $(λ_i)_{1\le i\le n}$ and the spectral radius $R_n=\max_{1\le i\le n}|λ_i|.$ Set $X_n=\sqrt{4 γ_{n}}(R_{n}-\sqrt{n}-\frac12\sqrt{γ_{n}})$ and $F_n$ be its distribution function, where $γ_{n}=\log n-2\log(\sqrt{2π}\log n).$ It was proved in \cite{Rider 2003} that $F_n$ converges weakly to the Gumbel distribution $Λ.$ We prove in further in this paper that $$\lim_{n\to\infty} \frac{\log n}{\log\log n}\, W_1\left(F_n, Λ\right)=2$$ and the Berry-Esseen bound $$\lim\limits_{n\to \infty} \frac{\log n}{\log\log n}\sup_{x\in \mathbb{R}}|F_{n}(x)-e^{-e^{-x}}|=\frac{2}{e}.$$