Interacting Loop Ensembles and Bose Gases

We study interacting Bose gases in thermal equilibrium on a lattice. We establish convergence of the grand-canonical Gibbs states of such gases to their mean-field (classical field) and large-mass (classical particle) limits. The former is a classical field theory for a complex scalar field with quartic self-interaction. The latter is a classical theory of point particles with two-body interactions. Our analysis is based on representations in terms of ensembles of interacting random loops, the Ginibre loop ensemble for Bose gases and the Symanzik loop ensemble for classical scalar field theories. For small enough interactions, our results also hold in infinite volume. The results of this paper were previously sketched in Fröhlich et al. (J Stat Phys 180(1–6):810–831, 2020).