On the probabilistic approach for Gaussian Berezin integrals

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.