  This paper studies probabilistic mean-field models for interacting bosons at
a positive temperature in the thermodynamic limit with random particle density.
In particular, we prove large deviation principles for empirical cycle counts
in all our models, and as such generalise recent work in \cite{ACK} where upper
and lower large deviation bounds do not match. Namely, on the one hand we
generalise to the so-called grand canonical ensemble, and on the other hand, we
consider classes of interaction potentials depending on the cycle counts which
are not restricted to be positive. Our large deviation results provide
representation formulae for the thermodynamic limit of the pressure which in
turn leads to proving Bose-Einstein-condensation (BEC) in all our models.
  A primary focus and novelty is the pressure representation via extended large
deviation analysis for the so-called hard-sphere, or HYL-model
(Huang-Yang-Luttinger) studied in \cite{Lew86}. This model has negative counter
terms in the Hamiltonian and shows BEC depending on the coupling constants.
