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.
Categories: math.PR