To illustrate the nature of the polymer around the transition, we present some typical configurations with specified numbers of contacts that have been generated in a simulation at \(\omega=0.5\). These can be seen in Figure \ref{fig:configs}. Not only do these configurations illustrate the nature of the low-temperature phase where the number of contacts is large (see the configuration with \(m_t=452\)), but they also clearly demonstrate the first-order nature of the collapse, as we observe co-existence of fully dense and swollen parts of the polymer for smaller values of \(m_t\).

Now let us consider low temperatures for fixed values of \(\tau\). In our previous work on INNSAT \cite{Bedini_2013_INNSAT}, when \(\tau=1\) the quantity \(p_n\) was seen to converge to a non-zero value regardless of temperature: see Figure 7 in \cite{Bedini_2013}. This phase is unambiguously of a different nature as that for large \(\tau\) at fixed \(\omega\). We can therefore conclude that the line of peaks for large values of \(\omega\) and \(\tau\) in the largest eigenvalue plot is associated with a transition between these two low temperature phases: one being maximally dense and the other not. We have not investigated this transition here but an analysis of a transition between similar phases \cite{Krawczyk_2009} leads us to conjecture that it is second order.