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

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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 (hereN 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 orderN. 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 areυ=0.7496±0.0007 ind=2 andυ= 0.592±0.003 ind=3 (95% confidence limits), based on SAWs of lengths 200⩽N⩽10000 and 200⩽N⩽ 3000, respectively.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, MSS 1A1, Toronto, Ontario, Canada

    Neal Madras

  2. Department of Physics, New York University, 10003, New York, New York

    Alan D. Sokal

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  1. Neal Madras
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  2. Alan D. Sokal
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Madras, N., Sokal, A.D. The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk. J Stat Phys 50, 109–186 (1988). https://doi.org/10.1007/BF01022990

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