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.
