Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles

TL;DR

This paper studies eigenvalue distributions of a family of non-Hermitian random matrices, revealing a universal interpolation between Airy and Poisson statistics at the spectral edge, and provides a simplified integral representation of the kernel.

Contribution

It demonstrates the universality of the interpolating eigenvalue kernel for different non-Hermitian ensembles and offers a new integral form highlighting its structure.

Findings

Eigenvalue statistics interpolate between Airy and Poisson near the spectral edge.

The interpolating kernel is universal across different ensembles.

A simplified integral representation of the kernel is provided.

Abstract

We consider a family of chiral non-Hermitian Gaussian random matrices in the unitarily invariant symmetry class. The eigenvalue distribution in this model is expressed in terms of Laguerre polynomials in the complex plane. These are orthogonal with respect to a non-Gaussian weight including a modified Bessel function of the second kind, and we give an elementary proof for this. In the large $n$ limit, the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between Airy and Poisson that was recently found by one of the authors for the elliptic Ginibre ensemble. We conclude that this scaling limit is universal, appearing for two different non-Hermitian random matrix ensembles with unitary symmetry. As a second result we give an equivalent form for the interpolating Airy kernel in terms of a single real integral, similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum. This makes its structure as a one-parameter deformation of the Airy kernel more transparent.