# A Faster Implementation of the Pivot Algorithm for Self-Avoiding Walks

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## 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.

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