Deviation and moderate deviation for extremal eigenvalues of large Chiral non-Hermitian random matrices

TL;DR

This paper investigates the probabilities of deviations for the largest and smallest eigenvalues' magnitudes in large chiral non-Hermitian random matrices, providing new probabilistic bounds for their spectral radii.

Contribution

It establishes deviation and moderate deviation probabilities for extremal eigenvalues of large chiral non-Hermitian random matrices, a novel probabilistic analysis in this context.

Findings

Derived deviation probabilities for spectral radius.

Established moderate deviation bounds.

Analyzed extremal eigenvalue behavior in large matrices.

Abstract

Consider the chiral non-Hermitian random matrix ensemble with parameters $n$ and $v,$ and let $(\zeta_i)_{1\le i\le n}$ be its $n$ eigenvalues with positive $x$-coordinate. In this paper, we establish deviation probabilities and moderate deviation probabilities for the spectral radius $(n/(n+v))^{1/2}\max_{1\le i\le n}|\zeta_i|^2,$ as well as $(n/(n+v))^{1/2}\min_{1\le i\le n}|\zeta_i|^2.$