The properties of lattice polymers are also related to those of magnetic systems near their critical point \cite{vanderzande1998a-a}. More precisely, self-avoiding walks configurations appear as the diagrams of the high-temperature expansion of an \(O(n)\) magnetic system when taking the formal limit of zero components (\(n \to 0\)), and their scaling exponents can be obtained from the \(O(n)\) critical point. In this mapping the collapse transition corresponds to a tri-critical point of the magnetic system and one would hope to obtain the critical exponents for the polymer collapse transition from the ones of this tri-critical point.
Various authors \cite{Nienhuis_1982,Nienhuis_1984,Blote_1989,Guo_2006,Nienhuis_2008} have studied critical and tri-critical \(O(n)\) spin systems. For a special choice of the model on the honeycomb lattice, exact results were obtained in \cite{Nienhuis_1982} for two cases: a critical point and a special point governing the low-temperature phase. When \(n \to 0\) these two cases become the dilute and dense polymer phase. The dilute and dense phases were also found along two branches of a square-lattice \(O(n)\) model \cite{Nienhuis_1984,Blote_1989} together with two different branches describing the critical behaviour that occurs when \(O(n)\) and Ising degrees of freedom on the square lattice display a joint critical point.
On the other hand, Duplantier and Saleur in 1987 \cite{Duplantier_1987} realised that, on the honeycomb lattice, an self-interaction for SAW could be obtained by introducing vacancies, hexagonal faces that the SAW is not allowed to touch. Using this observation they could obtain a set of critical exponents for the polymer collapse transition which have been subsequently found to correctly describe the collapse in the ISAW model (see the extensive list of references in \cite{Caracciolo_2011} for example). We will refer to the universality class of this critical point as the ‘\(\theta\)-point’. An exact description has now been proposed \cite{Guo_2006,Nienhuis_2008} for the tri-critical \(O(n)\) model in two dimensions as a function of \(n\).
When it comes to ISAT the scenario is much less clear, in particular it not obvious how the change of topology caused by the presence of crossings affects the above picture. The description in terms of height model and Coulomb Gas allows one to consider the presence of crossings only as a perturbation (see \cite{Jacobsen_2009} for a review of these methods). The exponent associated to loop crossings is the same as that of cubic symmetry breaking, which is known to be irrelevant in the critical \(O(n)\) phase, but it has been observed \cite{Jacobsen_2003} that this is not true in the low-temperature phase, where the introduction of crossings is a relevant operator which leads to a different universality class. This is generically referred to as the Goldstone phase and it is believed \cite{Jacobsen_2003,Nienhuis_2008} to be described by the intersecting loop model proposed in \cite{Martins_1998,Martins_1998} and since called the Brauer model \cite{Gier_2005}.
The relevance of crossings at the collapse point is not clear. While the cubic perturbation is still believed to be relevant at the tri-critical \(O(n)\) point \cite{Nienhuis_2008}, a recent numerical study \cite{Bedini_2013_INNSAT} seems to indicate that the Duplantier and Saleur universality class is stable in the presence of crossings, at least with respect to the cross-over and length-scale exponents.
Very recently Nahum et al. \cite{Nahum_2013} published a study of loop models with crossings. Their analysis is based on a replica limit of the \(\sigma\) model on real projective space \(\mathbb{RP}^{n-1}\). They give a field theoretic description of the ISAT which explains the phase diagram found numerically in \cite{Foster_2009} and suggests that the ISAT collapse transition is an infinite-order multi-critical point.