Arbitrary unitarily invariant random matrix ensembles and supersymmetry

We generalize the supersymmetry method in random matrix theory to ensembles which are unitarily invariant, but otherwise arbitrary. Our exact approach extends a previous contribution in which we constructed a supersymmetric representation for the class of norm-dependent random matrix ensembles. Here, we derive a supersymmetric formulation under very general circumstances. A reduced probability density and a projector are identified that map the probability density from ordinary to superspace. Furthermore, it is demonstrated that setting up the theory in Fourier superspace has considerable advantages. General and exact expressions for the correlation functions are given. We also show how the use of hyperbolic symmetry can be circumvented in the present context in which the nonlinear σ model is not used. We construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the Green's functions.