On three-dimensional self-avoiding walk symmetry classes

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In two dimensions the universality classes of self-avoiding walks (SAWs) on the square lattice, restricted by allowing only certain two-step configurations (TSCs) to occur within each walk, has been argued to be determined primarily by the symmetry of the set of allowed rules. In three dimensions early work tentatively found one (undirected) universality class different to that of unrestricted SAWs on the simple cubic lattice. This rule was a natural generalization of the square lattice `spiral' SAW to three dimensions. In this report we examine a variety of three-dimensional SAW models with different step restrictions, carefully chosen so as to search for a connection between the symmetry of the rules and possible new universality classes. A first analysis of the scaling of the radius of gyration suggests several universality classes, including the one found earlier, and perhaps some novel class(es). However, a classification of these universality classes using the symmetries of the rules, or other basic rule properties, is not evident. Further analysis of the number of configurations and moment of inertia tensor suggests that in three dimensions the only non-trivial or undirected universality class is that of unrestricted SAWs.