Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution

Abstract

Let $X$ be a real $(\beta=1)$ or complex $(\beta=2)$ Ginibre ensemble. Let $\{\sigma_i\}_{1\le i\le n}$ be the eigenvalues of $X,$ and $Z_n$ be some rescaled version of $\max_i \Re \sigma_i.$ It was proved that $Z_n$ converges weakly to the Gumbel distribution $\Lambda_{\beta}$ with distribution function $e^{-\frac{\beta}{2}e^{-x}}.$ We further prove that $$\sup_{x\in \mathbb{R}}|\mathbb{P}(Z_n \leq x)-e^{-\frac{\beta}{2}e^{-x}}|=\frac{25\log \log n}{4e \log n}(1+o(1))$$ and $$ W_1\left(\mathcal{L}(Z_n), \Lambda_{\beta}\right)=\frac{25\log \log n}{4\log n}(1+o(1))$$ for sufficiently large $n$, where $\mathcal{L}(Z_n)$ is the distribution of $Z_n$ and $W_1$ is the Wasserstein distance. Similar results hold for $\max_{i} |\sigma_i|.$ Furthermore, the convergence rates of the complex Ginibre ensemble are universal for complex iid random matrices under certain moment conditions on entries.