Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution

TL;DR

This paper establishes precise convergence rates for the maximum real part of eigenvalues of Ginibre ensembles to the Gumbel distribution, demonstrating universality for complex iid matrices under certain conditions.

Contribution

It provides explicit convergence rate bounds for the eigenvalue extremal distribution and proves universality across complex iid matrices with moment conditions.

Findings

Convergence rate of order (log log n)/log n for eigenvalue maxima

Explicit bounds on distributional and Wasserstein distances

Universality of convergence rates for complex iid matrices

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.