Enumeration of self-avoiding walks on the square lattice

TL;DR

This paper introduces a new parallel algorithm for enumerating self-avoiding walks on the square lattice, extending enumeration to length 71 and analyzing critical exponents with high precision.

Contribution

The paper presents a novel parallel enumeration algorithm and provides extensive series data for self-avoiding walks, confirming theoretical predictions of critical exponents.

Findings

Enumerated walks up to length 71 using 128 processors.

Derived series for metric properties up to length 59.

Confirmed critical exponents $b3$ and $bd$ match exact values.

Abstract

We describe a new algorithm for the enumeration of self-avoiding walks on the square lattice. Using up to 128 processors on a HP Alpha server cluster we have enumerated the number of self-avoiding walks on the square lattice to length 71. Series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and mean-square distance of monomers from the end points have been derived to length 59. Analysis of the resulting series yields accurate estimates of the critical exponents $\gamma$ and $\nu$ confirming predictions of their exact values. Likewise we obtain accurate amplitude estimates yielding precise values for certain universal amplitude combinations. Finally we report on an analysis giving compelling evidence that the leading non-analytic correction-to-scaling exponent $\Delta_1=3/2$.