The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk

Abstract

The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (here N is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one "effectively independent" sample can be produced in a computer time of order N. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents are υ=0.7496±0.0007 in d=2 and υ= 0.592±0.003 in d=3 (95% confidence limits), based on SAWs of lengths 200≤N≤10000 and 200≤N≤ 3000, respectively. © 1988 Plenum Publishing Corporation.