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

## TL;DR

This paper derives new scaling laws for self-avoiding walks, specifically bridges and worms, relating their critical exponents to known exponents, supported by numerical evidence in 2D and 3D.

## Contribution

The paper introduces novel scaling relations for self-avoiding bridges and worms, connecting their critical exponents to existing exponents, with numerical validation.

## Key findings

- Derived the relation b3_b = b3_{1,1} +
u.
- Established b3_w = b3 -
u.
- Numerical evidence supports these relations in 2D and 3D.

## 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.