  We consider a model of $d$-dimensional interacting quantum Bose gas,
expressed in terms of an ensemble of interacting Brownian bridges in a large
box and undergoing the influence of all the interactions between the legs of
each of the Brownian bridges. We study the thermodynamic limit of the system
and give an explicit formula for the limiting free energy and a necessary and
sufficient criterion for the occurrence of a condensation phase transition. For
$d\geq 5$ and sufficiently small interaction, we prove that the condensate
phase is not empty. The ideas of proof rely on the similarity of the
interaction to that of the self-repellent random walk, and build on a lace
expansion method conducive to treating {\it paths} undergoing mutual repellence
within each bridge.
