Optimal $W_1$ and Berry-Esseen bound between the spectral radius of large Chiral non-Hermitian random matrices and Gumbel

Abstract

Consider the chiral non-Hermitian random matrix ensemble with parameters $n$ and $v$ and the non Hermiticity parameter $\tau =0$ and let $({\zeta }_i{)}_{1\le i\le n}$ be its $n$ eigenvalues with positive $x$-coordinate. Set $$X_n:=\sqrt{\log s_n}\left(\frac{2n{\max }_{1\le i\le n}|{\zeta }_i{|}^2-2\sqrt{n(n+v)}}{\sqrt{2n+v}}-a(s_n)\right)$$ with $s_n=n(n+v)/(2n+v)$ and $a(s_n)=\sqrt{\log s_n}-\frac{\log (\sqrt{2\pi }\log s_n)}{\sqrt{\log s_n}}.$ It was proved in \cite{JQ} that $X_n$ converges weakly to the Gumbel distribution $\Lambda$. In this paper, we give in further that $$\underset{n\to \infty }{\lim }\frac{\log s_n}{(\log \log s_n{)}^2}{W}_1\left(F_n,\Lambda \right)=\frac{1}{2}$$ and the Berry-Esseen bound $$\underset{n\to \infty }{\lim }\frac{\log s_n}{(\log \log s_n{)}^2}\underset{x\in \mathbb{R}}{\sup }|F_n(x)-e^{-e^{-x}}|=\frac{1}{2e}.$$ Here, $F_n$ is the distribution (function) of $X_n.$

25 pp