Large deviation analysis for classes of interacting Bosonic cycle counts

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.