  The interacting quantum Bose gas is a random ensemble of many Brownian
bridges (cycles) of various lengths with interactions between any pair of legs
of the cycles. It is one of the standard mathematical models in which a proof
for the famous Bose-Einstein condensation phase transition is sought for. We
introduce a simplified version of the model with an organisation of the
particles in deterministic boxes instead of Brownian cycles as the marks of a
reference Poisson point process (for simplicity, in $\mathbb Z^d$ instead of
$\mathbb R^d$).
  We derive an explicit and interpretable variational formula in the
thermodynamic limit for the limiting free energy of the canonical ensemble for
any value of the particle density. This formula features all relevant physical
quantities of the model, like the microscopic and the macroscopic particle
densities, together with their mutual and self-energies and their entropies.
The proof method comprises a two-step large-deviation approach for marked
Poisson point processes and an explicit distinction into small and large marks.
  In the characteristic formula, each of the microscopic particles and the
statistics of the macroscopic part of the configuration are seen explicitly;
the latter receives the interpretation of the condensate. The formula enables
us to prove a number of properties of the limiting free energy as a function of
the particle density, like differentiability and explicit upper and lower
bounds, and a qualitative picture below and above the critical threshold (if it
is finite). This proves a modified saturation nature of the phase transition.
However, we have not yet succeeded in proving the existence of this phase
transition.
