On correlation functions of characteristic polynomials for chiral Gaussian Unitary Ensemble

TL;DR

This paper derives a general spectral correlation function for products and ratios of characteristic polynomials in the chiral Gaussian Unitary Ensemble, using advanced integral techniques and providing asymptotic formulas for large matrices.

Contribution

It introduces a new integral representation for correlation functions in chGUE and generalizes existing formulas to include ratios of characteristic polynomials.

Findings

Correlation functions expressed as determinants.

Derived asymptotic formulas for large matrices.

Extended known results to ratios of polynomials.

Abstract

We calculate a general spectral correlation function of products and ratios of characteristic polynomials for a $N\times N$ random matrix taken from the chiral Gaussian Unitary Ensemble (chGUE). Our derivation is based upon finding an Itzykson-Zuber type integral for matrices from the non-compact manifold ${\sf{Gl(n,{\mathcal{C}})/U(1)\times ...\times U(1)}}$ (matrix Macdonald function). The correlation function is shown to be always represented in a determinant form generalising the known expressions for only positive moments. Finally, we present the asymptotic formula for the correlation function in the large matrix size limit.