Self-avoiding walks and polygons on the triangular lattice

TL;DR

This paper extends the enumeration of self-avoiding walks and polygons on the triangular lattice to lengths 40 and 60 using new algorithms, providing precise estimates for the connective constant and confirming theoretical predictions.

Contribution

Introduces new algorithms based on the finite lattice method to extend enumeration lengths and compute series for various properties of self-avoiding walks and polygons.

Findings

Estimated the connective constant as 4.150797226(26).

Confirmed theoretical predictions for critical exponents.

Provided series for metric properties and moments of area.

Abstract

We use new algorithms, based on the finite lattice method of series expansion, to extend the enumeration of self-avoiding walks and polygons on the triangular lattice to length 40 and 60, respectively. For self-avoiding walks to length 40 we also calculate series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and the mean-square distance of a monomer from the end points. For self-avoiding polygons to length 58 we calculate series for the mean-square radius of gyration and the first 10 moments of the area. Analysis of the series yields accurate estimates for the connective constant of triangular self-avoiding walks, $\mu=4.150797226(26)$, and confirms to a high degree of accuracy several theoretical predictions for universal critical exponents and amplitude combinations.