A novel formula for Gaussian Berezin integrals

We present a novel formula for 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. Our formula allows one to evaluate these integrals without calculating the inverse of the matrix. The derivation of the formula is obtained via a recently proposed method to calculate Berezin integrals as an expectation of suitable functionals of Poisson processes. The comparison of our formula with the old one gives rise to an identity which has interesting algebraic, geometrical and physical aspects. In terms of cost of calculation, the gain obtained using our formula with respect to the old one is rapidly increasing with the dimension of the Grassmann algebra and the order of correlation. Furthermore, we have a mapping between two fermionic systems, not necessarily Gaussian, with short and long range interaction, respectively.