  We present a novel approach to Gaussian Berezin correlation functions. A
formula well known in the literature expresses these quantities in terms of
submatrices of the inverse matrix appearing in the Gaussian action. By using a
recently proposed method to calculate Berezin integrals as an expectation of
suitable functionals of Poisson processes, we obtain an alternative formula
which allows one to skip the calculation of the inverse of the matrix. This
formula, previously derived using different approaches (in particular by means
of the Jacobi identity for the compound matrices), has computational advantages
which grow rapidly with the dimension of the Grassmann algebra and the order of
correlation. By using this alternative formula, we establish a mapping between
two fermionic systems, not necessarily Gaussian, with short and long range
interaction, respectively.
