  There are several methods to treat ensembles of random matrices in symmetric
spaces, circular matrices, chiral matrices and others. Orthogonal polynomials
and the supersymmetry method are particular powerful techniques. Here, we
present a new approach to calculate averages over ratios of characteristic
polynomials. At first sight paradoxically, one can coin our approach
"supersymmetry without supersymmetry" because we use structures from
supersymmetry without actually mapping onto superspaces. We address two kinds
of integrals which cover a wide range of applications for random matrix
ensembles. For probability densities factorizing in the eigenvalues we find
determinantal structures in a unifying way. As a new application we derive an
expression for the k-point correlation function of an arbitrary rotation
invariant probability density over the Hermitian matrices in the presence of an
external field.
