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.

arXiv comment13 pages, 5 figures. Revised version expands on applicability of these results and adds many references. Dedicated to the memory of Vladimir Rittenberg

References in corpus (13)

• Accurate estimate of the critical exponent $\nu$ for self-avoiding walks via a fast implementation of the pivot algorithm
• Boundary and Interface CFTs from the Conformal Bootstrap
• Massive Field-Theory Approach to Surface Critical Behavior in Three-Dimensional Systems
• Simulations of grafted polymers in a good solvent
• Exact Results for the Adsorption of a Flexible Self-Avoiding Polymer Chain in Two Dimensions
• Surface critical behavior in fixed dimensions $d<4$: Nonanalyticity of critical surface enhancement and massive field theory approach
• Scale-free Monte Carlo method for calculating the critical exponent $\gamma$ of self-avoiding walks
• Copolymer Networks and Stars: Scaling Exponents
• The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is $1+\sqrt{2}$
• The self-avoiding walk in a strip
• Three-dimensional terminally attached self-avoiding walks and bridges
• Exact Theory of Polymer Adsorption in Analogy with the Kondo Problem
• Two-dimensional polymer networks at a mixed boundary: Surface and wedge exponents

Cited by in corpus (3)

• Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents
• Statistical Mechanics of Confined Polymer Networks
• On the existence of critical exponents for self-avoiding walks