On the probabilistic approach for Gaussian Berezin integrals

TL;DR

This paper introduces a new computational method for Gaussian Berezin integrals that avoids matrix inversion, offering efficiency improvements for high-dimensional problems and enabling mappings between different fermionic systems.

Contribution

It presents an alternative formula for Gaussian Berezin integrals using Poisson process expectations, bypassing inverse matrix calculations and enhancing computational efficiency.

Findings

New formula reduces computational complexity for high-dimensional integrals

Enables mapping between different fermionic systems with various interaction ranges

Demonstrates advantages over traditional inverse-based methods

Abstract

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.