On the Non-Universality of a Critical Exponent for Self-Avoiding Walks

TL;DR

This paper extends enumeration data for self-avoiding walks on the Manhattan lattice, providing numerical evidence that the critical exponent gamma varies with lattice topology, challenging the notion of universality.

Contribution

It offers new enumeration data and supports the idea that the critical exponent gamma is non-universal across different lattice types.

Findings

Estimated gamma = 1.3385 ± 0.003 for Manhattan lattice

Exponent differs from that on square, triangular, and honeycomb lattices

Lattice topology influences the critical exponent gamma

Abstract

We have extended the enumeration of self-avoiding walks on the Manhattan lattice from 28 to 53 steps and for self-avoiding polygons from 48 to 84 steps. Analysis of this data suggests that the walk generating function exponent gamma = 1.3385 +- 0.003, which is different from the corresponding exponent on the square, triangular and honeycomb lattices. This provides numerical support for an argument recently advanced by Cardy, to the effect that excluding walks with parallel nearest-neighbour steps should cause a change in the exponent gamma. The lattice topology of the Manhattan lattice precludes such parallel steps.