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 $$\underset{x\in \mathbb{R}}{\sup }|\mathbb{P}(Z_n\le x)-e^{-\frac{\beta }{2}{e}^{-x}}|=\frac{25\log \log n}{4e\log n}(1+o(1))$$ 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.