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

TL;DR

This paper precisely characterizes the convergence rate of the spectral radius distribution of the complex Ginibre ensemble to the Gumbel distribution, providing exact asymptotic bounds and convergence metrics.

Contribution

It establishes the exact asymptotic convergence rate of the spectral radius distribution to the Gumbel law, including Wasserstein and Berry-Esseen bounds.

Findings

Wasserstein distance scaled limit equals 2

Berry-Esseen bound scaled limit equals 2/e

Convergence to Gumbel distribution with explicit rate

Abstract

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