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

TL;DR

This paper studies the spectral radius of large chiral non-Hermitian random matrices, establishing optimal bounds for the Wasserstein-1 distance and Berry-Esseen bounds between its distribution and the Gumbel distribution.

Contribution

It provides the first precise asymptotic bounds for the convergence rate of the spectral radius distribution to the Gumbel law in this matrix ensemble.

Findings

W1 distance scaled limit is 1/2

Berry-Esseen bound scaled limit is 1/(2e)

Convergence to Gumbel distribution established

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 $$\lim_{n\to\infty} \frac{\log s_n}{(\log\log s_n)^2}W_1\left(F_n, \Lambda\right)=\frac{1}{2}$$ 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 $X_n.$