Efficient implementation of the pivot algorithm for self-avoiding walks

TL;DR

This paper presents a significantly faster implementation of the pivot algorithm for simulating long self-avoiding walks, with data structures and heuristics that enable efficient large-scale computations on lattice models.

Contribution

The authors introduce an optimized implementation of the pivot algorithm, achieving near-constant mean time per attempted pivot for very long walks, and demonstrate its effectiveness through extensive numerical experiments.

Findings

Mean time per pivot is approximately O(1) for large N.

Numerical results show sub-logarithmic to logarithmic scaling of computation time.

The method is adaptable to various polymer models and lattice types.

Abstract

The pivot algorithm for self-avoiding walks has been implemented in a manner which is dramatically faster than previous implementations, enabling extremely long walks to be efficiently simulated. We explicitly describe the data structures and algorithms used, and provide a heuristic argument that the mean time per attempted pivot for $N$-step self-avoiding walks is $O(1)$ for the square and simple cubic lattices. Numerical experiments conducted for self-avoiding walks with up to 268 million steps are consistent with $o(\log N)$ behavior for the square lattice and $O(\log N)$ behavior for the simple cubic lattice. Our method can be adapted to other models of polymers with short-range interactions, on the lattice or in the continuum, and hence promises to be widely useful.