On three-dimensional self-avoiding walks

Long self-avoiding walks of up to 2400 steps have been generated on the SC and BCC lattices using an improved Monte Carlo technique. Analysis of the asymptotic length dependence of the end-to-end distance and radius of gyration of the walks leads to values in the range 0.591-0.593 for the critical exponent nu . The simple power-law dependence on length provides an excellent fit to the data over walk lengths between 120 and 2400 with fluctuations around the asymptotic results averaging to a mere 0.13%. Systematic deviations from asymptotic behaviour in short self-avoiding walks have also been examined using Monte Carlo generated walks ranging in length from 12 to 60; the results do not support the existence of the non-analytic correction predicted by the renormalisation group. In the light of this unexpected result, available series expansions for walks on the FCC lattice have been re-examined and previous claims to have observed non-analytic behaviour questioned; equal, if not better convergence of the extrapolated series can be obtained without resorting to non-analytic correction terms. Finally, an analysis has been made of the radius of gyration series to which several new terms have been added.