Deviation probabilities and Sharp Berry-Esseen bound for rightmost eigenvalue of large non-Hermitian chiral random matrices

This paper provides a quantitative analysis of the rightmost eigenvalue for a chiral non-Hermitian random Dirac matrix in the maximally non-Hermitian regime ($\tau =0$). Let $({\sigma }_i{)}_{1\le i\le n}$ be the eigenvalues with positive real part. We define the normalization constants $$s_n=\frac{4n(n+v)}{2n+v},{\gamma }_n=\frac{1}{2}\log s_n-\frac{5}{4}\log (\log s_n)-\log \left(2^{1\mathrm{/}4}\pi \right),$$ and the centered and scaled variable $$X_n=\sqrt{2s_n\log s_n}\left(\right(\frac{n}{n+v}{)}^{1\mathrm{/}4}\underset{1\le i\le n}{\max }\mathrm{\Re }{\sigma }_i-1-\frac{{\gamma }_n}{\sqrt{2s_n\log s_n}})\mathrm{.}$$ Our main result is the following sharp Berry–Esseen bound for the convergence of $X_n$ to the Gumbel distribution: $$\underset{x\in \mathbb{R}}{\sup }\mid \mathbb{P}(X_n\le x)-e^{-e^{-x}}\mid =\frac{25(\log \log s_n{)}^2}{16e\log s_n}(1+o(1)),$$ which holds as $n\to \mathrm{\infty }$ for an arbitrary parameter $v\ge 0$ (which may depend on $n$). As a byproduct of our analysis, we also obtain precise large- and moderate-deviation principles for the scaled rightmost eigenvalue $(\frac{n}{n+v}{)}^{1\mathrm{/}4}{\max }_{1\le i\le n}\mathrm{\Re }{\sigma }_i$, characterizing its rate of convergence to the value $1$.