Optimal W₁ and Berry-Esseen bound between the spectral radius of large Chiral non-Hermitian random matrices and Gumbel

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 $$\lim_{n\to\infty} \frac{\log s_n}{(\log\log s_n)^2}W_1\left(F_n, \Lambda\right)=\frac{1}{2}$$ and the Berry-Esseen bound $$\lim_{n\to\infty} \frac{\log s_n}{(\log\log s_n)^2}\sup_{x\in\mathbb{R}}|F_n(x)-e^{-e^{-x}}|=\frac{1}{2e}.$$ Here, $F_n$ is the distribution (function) of $X_n.$