Exact Enumeration Study of Free Energies of Interacting Polygons and Walks in Two Dimensions

TL;DR

This study uses extended series analysis to compare free energies of interacting polygons and walks in two dimensions, providing evidence they are equal above the theta-temperature and exploring correction-to-scaling effects.

Contribution

The paper extends series data for ISAW and ISAP, demonstrating their free energies are equal above the theta-temperature and analyzing correction-to-scaling effects in two dimensions.

Findings

Free energies of ISAW and ISAP are equal above the theta-temperature.

Differences below the theta-temperature are due to correction-to-scaling.

Analysis of neighbor-avoiding SAW series is also presented.

Abstract

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, 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 to at least within 1%. 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 cross-over exponent previously estimated for ISAP to that predicted theoretically, and numerically confirmed, for ISAW. An analysis of newly extended neighbour-avoiding SAW series is also given.