On characteristic polynomials for a generalized chiral random matrix ensemble with a source

TL;DR

This paper analyzes characteristic polynomials in a generalized chiral random matrix ensemble with an external source, revealing asymptotic behaviors and kernels related to Bessel and Macdonald functions.

Contribution

It extends the study of characteristic polynomial averages to a deformed chiral ensemble with an external source, connecting to eigenvector statistics and asymptotic kernel analysis.

Findings

Derived asymptotics of characteristic polynomial averages in the ensemble.

Identified the kernel related to Bessel and Macdonald functions in the scaling limit.

Connected the results to eigenvector statistics in the complex Ginibre ensemble.

Abstract

We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a $N\times N$ random matrix taken from a $L$-deformed Chiral Gaussian Unitary Ensemble with an external source $\Omega$. Relation to a recently studied statistics of bi-orthogonal eigenvectors in the complex Ginibre ensemble, see Y.V. Fyodorov arXiv:1710.04699, is briefly discussed as a motivation to study asymptotics of these objects in the case of external source proportional to the identity matrix. In particular, for an associated 'complex bulk/chiral edge' scaling regime we retrieve the kernel related to Bessel/Macdonald functions.