Considering again the density plot of the largest eigenvalue, the line of peaks that is associated with first-order transitions for \(\omega <1\) and that meets the ISAT critical point for \(\omega =1\) extends to higher values of \(\omega\). This implies some type of phase transition at low temperatures. To understand what this transition might be, we now consider the low temperature phases for fixed values of \(\omega\) and \(\tau\). When \(\omega=1\), an analysis of the ISAT model has previously shown that the low temperature phase is maximally dense with the proportion \(p_n\) of sites on the trail that are not doubly occupied going to zero in the thermodynamic limit of infinite length \cite{Doukas_2010} \cite{Bedini_2013}. Here we plot the same quantity for various temperatures when \(\omega =0.5\) in Figure \ref{fig:low-temperature}. At low values of \(\tau\) the quantity \(p_n\) converges to a non-zero value while for larger values \(p_n\) seems to converge to zero within error, with a transition visible around \(\tau\approx2.5\).