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

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66 Scopus citations


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(Nq) 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(Nq) make an accurate estimate of q impossible. They also imply that the asymptotic behavior of O(Nq) 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.

Original languageEnglish (US)
Pages (from-to)407-429
Number of pages23
JournalJournal of Statistical Physics
Issue number3-4
StatePublished - 2002


  • Pivot algorithm
  • Polymer
  • Self-avoiding walk

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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