The statistics of collapsing square lattice trails with a fixed number of vertices of degree 4

A trail on the square lattice with a fixed number, k, of vertices of degree 4 is called a k-trail. We model polymer collapse using k-trails by incorporating an interaction energy which is proportional to the number of nearest-neighbour contact edges of the trail. It is known that the number of square lattice n-edge closed (open) k-trails can be bounded above and below (to O(nk)) by the number of n-step self-avoiding circuits (walks). This along with pattern theorems for self-interacting self-avoiding circuits and walks are used herein to establish upper and lower bounds (to O(nk)) for the collapsing free energy of k-trails in terms of self-avoiding circuits or walks, as appropriate. We also use pattern theorems to obtain bounds on the limiting nearest-neighbour contact density for collapsing k-trails. Finally, we investigate k-trails with a fixed density of nearest-neighbour contacts and show that their limiting entropy per monomer is independent of k