So far we have studied SAWs in the low fugacity regime where z ≤ μ-1. When z > μ-1, the grand partition function diverges. We can still give a meaning to it by making an analytic continuation. On the other hand, we know that SAWs appear in the high temperature expansion of the O(n)-model and that μ-l corresponds to the critical temperature of that model. The regime where z > μ-1 therefore corresponds to the low temperature phase of the O(n)-model. A spin model has of course a well defined low temperature regime and one might ask what this phase means for polymers. It is these questions which we study in the present chapter. The polymers in this phase are usually referred to as dense polymers. Our discussion will mostly be limited to the two-dimensional case.

The low temperature region of the O(n)-model

The study of polymers in the high fugacity regime can be performed in essentially two ways. The first way is to study the low temperature properties of the O(n)-model. This will be done using the techniques known from previous chapters; the Coulomb gas, exact solutions using the Bethe Ansatz, and so on. On the other hand we can study immediately the properties of walks themselves, in the regime where z > μ-1. Sure enough, in that region the grand partition function diverges, but the trick is to study the properties of walks in finite systems, e.g. in a finite box of volume Λ. A typical size for such a volume is Λ1/d. The finite volume leads to a cutoff for the grand partition sum.