Universality of the convergence rate for spectral radius of complex IID random matrices

TL;DR

This paper proves that the convergence rate of the spectral radius distribution of complex IID random matrices to the Gumbel distribution is universal, extending previous results from Ginibre ensembles to a broader class.

Contribution

It establishes the universality of the convergence rate of spectral radius distributions for complex IID matrices, generalizing prior results specific to Ginibre ensembles.

Findings

Convergence rate of spectral radius distribution is universal across complex IID matrices.

Explicit asymptotic formulas for the Kolmogorov and Wasserstein distances.

Extends prior results from Ginibre ensembles to more general IID matrices.

Abstract

Let $X$ be an $n\times n$ matrix with independent and identically distributed entries $x_{ij} \stackrel{\text { d }}{=} n^{-1 / 2} x$ for some complex random variable $x$ of mean zero and variance one. Let $\{\sigma_i\}_{1\le i\le n}$ be the eigenvalues of $X$ and let $|\sigma_1|:=\max_{1\le i\le n}|\sigma_i|$ be the spectral radius. Set $Y_n=\sqrt{4 n \gamma_n}\left[|\sigma_1|-1-\sqrt{\frac{\gamma_n}{4 n}}\right],$ where $\gamma_{n}=\log{n}-2\log{\log{n}}-\log{2\pi}.$ As established in \cite{Cipolloni23Universality}, with specific moment-related conditions imposed on $x,$ the Gumbel distribution $\Lambda$ is identified as the universal weak limit of $Y_n.$ Subsequently, we extend this line of research and rigorously prove that the convergence rate, previously obtained for complex Ginibre ensembles in \cite{MaMeng25}, also possesses the property of universality. Precisely, one gets $$\sup_{x\in \mathbb{R}}|\mathbb{P}(Y_n \leq x)-e^{-e^{-x}}|=\frac{2\log\log n}{e\log n}(1+o(1))$$ and $$W_1\left(\mathcal{L}(Y_n), \Lambda\right)=\frac{2\log\log n}{\log n}(1+o(1))$$ for sufficiently large $n$, where $\mathcal{L}(Y_n)$ is the distribution of $Y_n$.