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Critical exponents, hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks

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Abstract

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2Δ 4 −γ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2Δ 4 −γ. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios <R 2 g >/<R 2 e >=0.14026±0.00007, <R 2 m >/<R 2 e >=0.43961±0.00034, and Ψ*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponentΔ 1=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy forΔ 1. Earlier Monte Carlo estimates ofv, which were ≈0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R 2 g >/<R 2 e >=0.1599±0.0002 and Ψ*=0.2471±0.0003; since Ψ*>0, hyperscaling holds. The approach to Ψ* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2Δ 4 −γ for two-dimensional SAWs.

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Authors and Affiliations

  1. Debt and Equity Markets Group, Merrill Lynch, World Financial Center, 10281-1315, New York, New York

    Bin Li & Alan D. Sokal

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

    Bin Li

  3. Department of Mathematics and Statistics, York University, M3J 1P3, North York, Ontario, Canada

    Neal Madras

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Li, B., Madras, N. & Sokal, A.D. Critical exponents, hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks. J Stat Phys 80, 661–754 (1995). https://doi.org/10.1007/BF02178552

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