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

Let $X$ be an $n\times n$ matrix with independent and identically distributed entries $x_{ij}{n}^{-1\mathrm{/}2}x$ 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 $\mathrm{\mid }{\sigma }_1\mathrm{\mid }:={\max }_{1\le i\le n}\mathrm{\mid }{\sigma }_i\mathrm{\mid }$ be the spectral radius. Set $Y_n=\sqrt{4n{\gamma }_n}[\mathrm{\mid }{\sigma }_1\mathrm{\mid }-1-\sqrt{\frac{{\gamma }_n}{4n}}],$ where ${\gamma }_n=\log n-2\log \log n-\log 2\pi \mathrm{.}$ As established in \cite{Cipolloni23Universality}, with specific moment-related conditions imposed on $x,$ the Gumbel distribution $\mathrm{\Lambda }$ is identified as the universal weak limit of $Y_n\mathrm{.}$ 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 $$\underset{x\in \mathbb{R}}{\sup }\mathrm{\mid }\mathbb{P}(Y_n\le x)-e^{-e^{-x}}\mathrm{\mid }=\frac{2\log \log n}{e\log n}(1+o(1))$$ and $$W_1(\mathcal{L}(Y_n),\mathrm{\Lambda })=\frac{2\log \log n}{\log n}(1+o(1))$$ for sufficiently large $n$, where $\mathcal{L}(Y_n)$ is the distribution of $Y_n$.