Spectrum of self-avoiding walk exponents

A short range interaction is incorporated into the self-avoiding walk (SAW) model of polymer chains by partitioning SAW's into equivalence classes of chain configurations having m nearest-neighbor contacts, and performing an energetically weighted averaging over these restricted SAW configurations. Surprisingly, there have been limited studies of the geometrical properties of 'contact-constrained' SAW configurations, which contrasts with the well studied unrestricted SAW"s. Accordingly, we generate Monte Carlo data for the total number of SAW configurations ${\mathrm{C}}_{\mathrm{n},\mathrm{m}}$ having a fixed number of contacts m for chains of length n on square and cubic lattices. Applications of the standard ratio method to the ${\mathrm{C}}_{\mathrm{n},\mathrm{m}}$ data shows a unique connectivity constant μ (NAW), corresponding to neighbor-avoiding walks (m=0), and a 'spectrum' of γ exponents which depend on the contact number m. The asymptotic scaling of the number of contact-constrained SAW's is found to be similar to the number of lattice animals and random plaquette surfaces having a fixed cyclomatic index c and genus g, respectively. The existence of this common structure is promising for the development of an analytic theory of interacting polymers and surfaces.