The end-point distribution of self-avoiding walks on a crystal lattice

A recursive relation connecting the numbers of self-avoiding walks on an arbitrary network with the numbers of certain related topologies is given, and is used to obtain exact end-point distributions of self-avoiding walks on a variety of Bravais lattices in two and three dimensions. The growth (proportional to n gamma ) of the mean-square end-to-end distance of such walks of n steps is re-examined. The authors estimate (i) for the triangular lattice: gamma =1.488+or-0.002; (ii) for the face-centred cubic lattice: gamma =1.20+or-0.01. The first estimate is lower than gamma =1.5 conjectured by some workers in this field; however, there are differences in assumptions of how the mean-square end-to-end distance ought to depend on n.