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 {\em bridges} that relates the critical exponent for bridges $γ_b$ to that of terminally-attached self-avoiding arches, $γ_{1,1},$ and the {correlation} length exponent $ν.$ We find $γ_b = γ_{1,1}+ν.$ 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 $γ_w,$ which is $γ_w=γ-ν.$ This too is supported by enumerative results in the two-dimensional case.

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