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.
It is here that Monte Carlo assumes the role of importance sampling: instead of computing the sum over all auxiliary-field configurations, it concentrates on the most critical ones, with a certain weight. In classical statistical physics, this weight is typically given by the (positive semi-definite) Boltzmann factor.