We calculate a general spectral correlation function of products and ratios of characteristic polynomials for a N × N random matrix taken from the chiral Gaussian Unitary Ensemble (chGUE). Our derivation is based upon finding a Itzykson-Zuber type integral for matrices from the noncompact manifold Gl(n, C)/U(1) x ⋯ x U(1) (matrix Macdonald function). The correlation function is shown to be always represented in a determinant form generalizing the known expressions for only positive moments. Finally, we present the asymptotic formula for the correlation function in the large matrix size limit.
Bibliographical noteFunding Information:
The first author (Y.V.F.) is grateful to Jac Verbaarschot for triggering his interest in the problem and useful correspondence. Important comments by Thomas Guhr and Tilo Wettig as well as useful communications with Gernot Akemann are much appreciated. This research was supported by EPSRC grant GR/13838/01 “Random matrices close to unitary or Hermitian”.