Exact enumeration study of free energies of interacting polygons and walks in two dimensions

We present analyses of substantially extended series for both interacting self-avoiding walks (ISAW) and polygons (ISAP) on the square lattice. We argue that these provide good evidence that the free energies of both linear and ring polymers are equal above the theta-temperature, thus extending the application of a theorem of Tesi et al to two dimensions. Below the theta-temperature the conditions of this theorem break down, in contradistinction to three dimensions, but an analysis of the ratio of the partition functions for ISAP and ISAW indicates that the free energies are in fact equal at all temperatures within 1% at least. Any perceived difference can be interpreted as the difference in the size of corrections to scaling in both problems. This may be used to explain the vastly different values of the crossover exponent previously estimated for ISAP to thai predicted theoretically, and numerically confirmed, for ISAW. An analysis of newly extended neighbour-avoiding self-avoiding walk series is also given.