  We compute the limiting measure for the Feynman loop representation of the
Bose gas for a non mean-field energy. As predicted in previous works, for high
densities the limiting measure gives positive weight to random interlacements,
indicating the quantum Bose--Einstein condensation. We prove that in many cases
there is a shift in the critical density compared to the free/mean-field case,
and that in these cases the density of the random interlacements has a
jump-discontinuity at the critical point.
