Polymer chain statistics and universality. I

A brief summary is given of the concept of universality in the theory of critical phenomena. The concept is applied to random walks and self-avoiding walks on lattices corresponding to the n=-2 and n=0 universality classes. The Domb-Joyce model of a random walk on a lattice with a delta function interaction of strength w is identified with crossover behaviour, w serving as a crossover parameter. Exact enumerations are undertaken of the mean-square end-to-end length (R N 2 ) for the Domb-Joyce model for a number of three-dimensional lattices. Using the smoothness postulate of Griffiths, estimates are obtained of the asymptotic behaviour of the expansion factor alpha 2 =(R N 2 )/N in the range 0.5<w<1. By combining these with exact virial coefficients for small w the range is extended to w=0. The two-parameter approximation which assumes that alpha 2 is a function of wN 1/2 is satisfied with maximum errors of 2 or 3%. The two-parameter function which has been the subject of much discussion by polymer theorists is estimated and an empirical formula is proposed.