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.

13 pages, 5 figures. Revised version expands on applicability of these results and adds many references. Dedicated to the memory of Vladimir Rittenberg