New scaling laws for self-avoiding walks: bridges and worms

We show how the theory of the critical behaviour of d-dimensional polymer networks gives a scaling relation for self-avoiding bridges that relates the critical exponent for bridges gamma(b) to that of terminally-attached self-avoiding arches, gamma(11), and the correlation length exponent nu. We find gamma(b) = gamma(11) + nu. In the case of the special transition, we find gamma(b)(sp) = 1 2 [gamma(11)(sp) + gamma(11)] + nu. We provide compelling numerical evidence for this result in both two- and three-dimensions. Another subset of SAWs, called worms, are defined as the subset of SAWs whose origin and end-point have the same x-coordinate. We give a scaling relation for the corresponding critical exponent gamma(w), which is gamma(w) = gamma - nu. This too is supported by enumerative results in the two-dimensional case.