A faster implementation of the pivot algorithm for self-avoiding walks

TL;DR

This paper presents a significantly faster implementation of the pivot algorithm for simulating self-avoiding walks, reducing computational complexity and improving efficiency over previous methods, especially in two and three dimensions.

Contribution

The authors introduce a new implementation of the pivot algorithm that achieves sublinear time per accepted pivot, outperforming standard hash table methods.

Findings

Time per accepted pivot is O(N^q) with q<1, estimated below 0.57 in 2D and 0.85 in 3D.

Implementation is up to 80 times faster in 2D and 7 times faster in 3D than standard methods.

Effective q in simulations is around 0.7 in 2D and 0.9 in 3D, indicating improved efficiency.

Abstract

The pivot algorithm is a Markov Chain Monte Carlo algorithm for simulating the self-avoiding walk. At each iteration a pivot which produces a global change in the walk is proposed. If the resulting walk is self-avoiding, the new walk is accepted; otherwise, it is rejected. Past implementations of the algorithm required a time O(N) per accepted pivot, where N is the number of steps in the walk. We show how to implement the algorithm so that the time required per accepted pivot is O(N^q) with q<1. We estimate that q is less than 0.57 in two dimensions, and less than 0.85 in three dimensions. Corrections to the O(N^q) make an accurate estimate of q impossible. They also imply that the asymptotic behavior of O(N^q) cannot be seen for walk lengths which can be simulated. In simulations the effective q is around 0.7 in two dimensions and 0.9 in three dimensions. Comparisons with simulations that use the standard implementation of the pivot algorithm using a hash table indicate that our implementation is faster by as much as a factor of 80 in two dimensions and as much as a factor of 7 in three dimensions. Our method does not require the use of a hash table and should also be applicable to the pivot algorithm for off-lattice models.