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

TL;DR

This work analyzes the extreme eigenvalue behavior in the complex Ginibre ensemble by focusing on the spectral radius and its normalized form $X_n$ that converges to the Gumbel law. The authors establish the exact convergence rate in Wasserstein-1 distance to the Gumbel distribution and a Berry-Esseen type bound for the convergence, identifying precise logarithmic scaling factors. The core method hinges on Kostlan's transformation to Gamma-type order statistics $Y_j$, allowing a detailed tail analysis of $Y_{(n)}$ via $u_n(k,x)$, Mills ratio, and Petrov's gamma results, followed by careful partitioning into regimes to derive the asymptotics of the associated integrals. The main contributions are the exact rate $\lim_{n\to\infty} \frac{\log n}{\log\log n} W_1(F_n,\Lambda)=2$ and the Berry-Esseen bound $\lim_{n\to \infty} \frac{\log n}{\log\log n}\sup_{x} |F_n(x)-e^{-e^{-x}}|=\frac{2}{e}$, providing sharp finite-sample corrections and confirming Gumbel universality with explicit constants for the complex case.

Abstract

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

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof