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

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 $\mathrm{\Omega }$. 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.