Optimal $W_1$ 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 $$X_n:=\sqrt{\log s_n}(\frac{2n\underset{1\le i\le n}{\max }\mathrm{\mid }{\zeta }_i{\mathrm{\mid }}^2-2\sqrt{n(n+v)}}{\sqrt{2n+v}}-a(s_n))$$ with $s_n=n(n+v)\mathrm{/}(2n+v)$ and $a(s_n)=\sqrt{\log s_n}-\frac{\log (\sqrt{2\pi }\log s_n)}{\sqrt{\log s_n}}\mathrm{.}$ It was proved in \cite{JQ} that $X_n$ converges weakly to the Gumbel distribution $\mathrm{\Lambda }$. In this paper, we give in further that $$\underset{n\to \mathrm{\infty }}{\lim }\frac{\log s_n}{(\log \log s_n{)}^2}{W}_1(F_n,\mathrm{\Lambda })=\frac{1}{2}$$ and the Berry-Esseen bound $$\underset{n\to \mathrm{\infty }}{\lim }\frac{\log s_n}{(\log \log s_n{)}^2}\underset{x\in \mathbb{R}}{\sup }\mathrm{\mid }{F}_n(x)-e^{-e^{-x}}\mathrm{\mid }=\frac{1}{2e}\mathrm{.}$$ Here, $F_n$ is the distribution (function) of $X_n\mathrm{.}$