Algebraic Techniques for Enumerating Self-Avoiding Walks on the Square Lattice

TL;DR

This paper introduces an algebraic method that significantly improves the enumeration of self-avoiding walks on a square lattice, enabling the analysis of longer walks and estimation of critical phenomena.

Contribution

The paper presents a novel algebraic approach that outperforms direct counting, allowing enumeration of longer walks and precise estimation of critical parameters.

Findings

Enumerated self-avoiding walks up to 39 steps.

Estimated critical point, exponent, and amplitude accurately.

Complexity of enumeration is reduced to approximately 3^{N/4} times polynomial.

Abstract

We describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of $N$ steps is of order $3^{N/4}$ times a polynomial in $N$, and so the approach is greatly superior to direct counting techniques. We have enumerated walks of up to 39 steps. As a consequence, we are able to accurately estimate the critical point, critical exponent, and critical amplitude.