On characteristic polynomials of random Hermitian matrices: Gaussian unitary ensemble and its chiral counterpart

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N×N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard ``supersymmetry'' approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard-Stratonovich transformation in the ``bosonic'' sector. The method, suggested recently by one of us, is shown to be capable of calculation when reinforced with a generalization of the Itzykson-Zuber integral to a non-compact integration manifold. The correlation function is shown to be always represented in a determinant form generalising the known expressions for only positive moments. The same method works successfully for the chiral counterpart of the GUE ensemble which is relevant for Quantum Chromodynamics and condensed matter problems.