Interacting self-avoiding walks and polygons in three dimensions

Self-interacting walks and polygons on the simple cubic lattice undergo a collapse transition at the -point. We consider self-avoiding walks and polygons with an additional interaction between pairs of vertices which are unit distance apart but not joined by an edge of the walk or polygon. We prove that these walks and polygons have the same limiting free energy if the interactions between nearest-neighbour vertices are repulsive. The attractive interaction regime is investigated using Monte Carlo methods, and we find evidence that the limiting free energies are also equal here. In particular, this means that these models have the same -point, in the asymptotic limit. The dimensions and shapes of walks and polygons are also examined as a function of the interaction strength.