There are various statistical mechanical models of the polymer collapse phase transition \cite{gennes1979a-a,vanderzande1998a-a}. However, there are two basic elements in each of the models: on the one \cite{gennes1979a-a} hand, the configurations of the polymer have some type of so-called excluded volume, that implies molecules are separate in space, and on the other hand the configurations have an attractive force between different parts of the polymer, that drives the transition. On lattices, both self-avoiding walks (SAW), where different sites of walk avoid being on the same site of the lattice, and self-avoiding trails (SAT), which are walks that can share sites though not bonds of the lattice, have been used as the configuration space for collapse models. The attractive force has been modelled both by adding energies for shared sites and also via so-called nearest-neighbour contacts, where sites adjacent on the lattice not joined by a step of the walk are given an energy. When SAW are coupled with nearest neighbour interactions the canonical ISAW model is reproduced (see \cite{vanderzande1998a-a} for a review), while when trails are coupled with site interactions the ISAT model \cite{Malakis_1976} is reproduced. Despite what one might expect from the principle of universality the collapse of these models appears to behave differently \cite{Shapir_1984,Owczarek_1995,Owczarek_2007}.
A recent study \cite{Bedini_2013_INNSAT} considered self-avoiding trails interacting via nearest-neighbour contacts (INNSAT) as a hybrid of the two models. Evidence from computer simulations showed that the collapse transition in INNSAT is different from the collapse transition in ISAT, which is a strong second order transition, but similar to ISAW which is predicted to be a weak second order transition where the specific heat converges at the transition. In ISAW collapse one needs to consider the third derivative of the free energy to see a divergent quantity and then only weakly divergent. It was also found that the low-temperature phase of the two trail-collapse models differ substantially: the phase associated with multiply visited site interactions is fully dense in the thermodynamic limit (as shown in \cite{Bedini_2013}). The low-temperature phase associated with nearest-neighbour contacts wasn’t fully dense as is believed to be the case for interacting self-avoiding walks \cite{Owczarek_1993}.