Field theoretic and Monte Carlo analysis of the Domb - Joyce model

Abstract

We present a field-theoretic analysis of high-precision Monte Carlo data for the Domb - Joyce model on the sc lattice. We vary the repulsion between two segments at the same point from zero (random walk) to infinity (self-avoiding walk). Eventually, we even include a repulsion between segments at neighbour points to increase the excluded volume beyond that of self-avoiding walks. The data for the end-to-end distance, the radius of gyration and the partition function clearly show the existence of two branches of universal behaviour. These two branches can be identified with the weak- and strong-coupling branch of the renormalization group, respectively. A quantitative analysis shows the ability of the standard field theoretic approach to describe the data, including the data for strong coupling, i.e. renormalized coupling u greater than its fixed point value . We conclude, in contrast with some claims in the literature, that the standard formalism of the renormalized field theory can be used even for (strong-coupling branch). In addition, exploiting the fast approach to asymptotic behaviour at the transition between weak and strong coupling, we obtain very precise estimates for the critical exponents of self-avoiding walks.