It is here, as in other related methods, that Monte Carlo enters the game in the guise of importance sampling: the large sum over auxiliary-field configurations is performed by sampling over the most important ones, with a certain probability. In classical statistical physics, this probability is usually given by the (positive semi-definite) Boltzmann factor.
In this specific scenario, Monte Carlo takes center stage in importance sampling: rather than evaluating the sum over all auxiliary-field configurations, it prioritizes the most significant ones, with a particular probability. In classical statistical physics, this probability is usually derived from the (positive semi-definite) Boltzmann factor.