\label{sec:isat-nn}
A model of interacting SAT with both nearest-neighbour interaction and site interactions can be defined as follows. Consider the set of bond-avoiding paths \(\mathcal T_n\) of length \(n\). Given a SAT \(\psi_n \in \mathcal T_n\), we associate an energy \(-\varepsilon_t\) every time the path visits the same site more than once, as in ISAT. Additionally, we define a contact whenever there is a pair of sites that are neighbours on the lattice but not consecutive on the walk, as in ISAW. We associate an energy \(-\varepsilon_c\) with each contact.
For each configuration \(\psi_n \in \mathcal T_n\) we count the number \(m_t(\psi_n)\) of doubly-visited sites and \(m_c(\psi_n)\) of contacts: see Figure \ref{fig:configuration}. Hence we associate with each configuration a Boltzmann weight \(\tau^{m_t(\psi_n)} \omega^{m_c(\psi_n)}\) where \(\tau = \exp(\beta \varepsilon_t)\), \(\omega = \exp(\beta \varepsilon_c)\), and \(\beta\) is the inverse temperature \(1/k_B T\). The partition function of the model is given by \[Z_n(\tau, \omega) = \sum_{\psi_n\in\mathcal T_n}\ \tau^{m_t(\psi_n)} \omega^{m_c(\psi_n)} .\] The probability of a configuration \(\psi_n\) is then \[p(\psi_n; \tau, \omega) = \frac{ \tau^{m_t(\psi_n)} \omega^{m_c(\psi_n)} }{ Z_n(\tau, \omega) } .\] The average of any quantity \(Q\) over the ensemble set of paths \(\mathcal T_n\) is given generically by \[\langle Q \rangle(n; \tau, \omega) = \sum_{\psi_n\in\mathcal T_n} Q(\psi_n) \, p(\psi_n; \tau, \omega) .\] In particular, we can define the average number of doubly-visited sites per site and their respective fluctuations as \[u^{(t)} = \frac{ \langle m_t \rangle }{n} \quad \mbox{ and } \quad c^{(t)} = \frac{ \langle m_t^2 \rangle - \langle m_t \rangle^2 }{n} .\] One can also consider the average number of contacts of the trail and their fluctuations \[u^{(c)} = \frac{ \langle m_c \rangle }{n} \quad \mbox{ and } \quad c^{(c)} = \frac{ \langle m_c^2 \rangle - \langle m_c \rangle^2 }{n} .\] This model interpolates between three previously-studied models; when we set \(\omega = 1\) the model reduces to the ISAT model. It should be noted that an important observation \cite{Doukas_2010,Bedini_2013} for ISAT is that the low temperature phase is maximally dense. On the square lattice this implies that if one considers the proportion of the sites on the trail that are at lattice sites which are not doubly occupied via \[p_n=\frac{n - 2 {\langle}m_t {\rangle}}{n}, \label{eq:p-n-def}\] then it is expected that \[p_n \rightarrow 0 \quad \mbox{as} \quad n\rightarrow \infty.\] Very recently the ISAT model has been studied in the context of a loop model with crossings \cite{Nahum_2013}, where it has been suggested that the ISAT collapse point is an infinite-order multi-critical point described by the \(O(n \to 1)\) sigma model studied in \cite{Jacobsen_2003}.
If otherwise we set \(\tau = 0\) doubly visited sites are excluded and the model reduces to the ISAW model. Finally if we set \(\tau = 1\) it becomes the INNSAT model studied in \cite{Bedini_2013_INNSAT}. One would think that the presence of crossings would affect the universality class of the collapse transition (e.g. as portrayed in \cite{Nahum_2013}) but in \cite{Bedini_2013_INNSAT} it was shown that the INNSAT model has a collapse transition in the same universality class as ISAW, that is the \(\theta\)-point.