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

Abstract

This paper provides a quantitative analysis of the rightmost eigenvalue for a chiral non-Hermitian random Dirac matrix in the maximally non-Hermitian regime ($\ddot{I}„=0$). Let $(\ddot{I}{ƒ}_i{)}_{1\le i\le n}$ be the eigenvalues with positive real part. We define the normalization constants \[ s_n = \frac{4n(n+v)}{2n+v}, \qquad γ_n = \frac{1}{2}\log s_n - \frac{5}{4}\log(\log s_n) - \log\bigl(2^{1/4}π\bigr), \] and the centered and scaled variable \[ X_n = \sqrt{2s_n\log s_n}\,\bigl(\bigl(\tfrac{n}{n+v}\bigr)^{1/4}\,\max_{1\le i\le n}\Reσ_i \;-\; 1 \;-\; \frac{γ_n}{\sqrt{2s_n\log s_n}}\bigr). \] Our main result is the following sharp Berry--Esseen bound for the convergence of $X_n$ to the Gumbel distribution: \[ \sup_{x \in \mathbb{R}} \bigl|\mathbb{P}(X_n \le x) - e^{-e^{-x}}\bigr| = \frac{25 (\log\log s_n)^2}{16 e \,\log s_n}\,\bigl(1 + o(1)\bigr), \] which holds as $n\to \infty$ 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 $(\frac{n}{n+v}{)}^{1/4}{\max }_{1\le i\le n}\Re \ddot{I}{ƒ}_i$, characterizing its rate of convergence to the value $1$.