On two-dimensional self-avoiding random walks

Following Nienhuis' (1982) exact evaluation of the connective constant of the honeycomb lattice self-avoiding walk model, and the exact exponent values he has conjectured, the author has re-examined the available data on all three regular two-dimensional lattices. In the case of the triangular lattice the author has additionally corrected and extended the extant series. The author finds support for Nienhuis' value gamma =1 11 / 32 for all three lattices, and further finds that alpha = 1 / 2 for all three lattices, as given by Nienhuis' result nu = 3 / 4 and the hyperscaling relation d nu =2- alpha . The author is unable to find any consistent evidence of a 'correction-to-scaling' exponent Delta 1 <1 from the walk generating function, though other workers have found such an exponent for the mean square end-to-end distance series. But an exponent Delta 1 >1 cannot be ruled out.