  In this paper, we study various models for random combinatorial partitions
using large deviation analysis for diverging scale of the reference process.
Scaling limits of similar models have been studied recently \cite{FSa,FSb}
going back to \cite{Ver96}. After studying the reference model, we provide a
complete analysis of two mean field models, one of which is well-know
\cite{BCMP05} and the other one is the cycle mean field model. Both models show
critical behaviour despite their rate functions having unique minimiser. The
main focus is then a model with negative counter term, the probabilistic
version of the so-called \emph{Huang-Yang-Luttinger} (HYL) model \cite{BLP}.
Criticality in this model is the existence of a critical parameter for which
two simultaneous minimiser exists. At criticality an order parameter is
introduced as the double limits for the density of cycles with diverging
length, and as such it extends recent work \cite{AD21}.
