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

Abstract: We show how the theory of the critical behaviour of $d$-dimensional polymer networks gives a scaling relation for self-avoiding {\em bridges} that relates the critical exponent for bridges $\gamma_b$ to that of terminally-attached self-avoiding arches, $\gamma_{1,1},$ and the {correlation} length exponent $\nu.$ We find $\gamma_b = \gamma_{1,1}+\nu.$ We provide compelling numerical evidence for this result in both two- and three-dimensions. Another subset of SAWs, called {\em 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.