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

Bin Li, Neal Madras, Alan D. Sokal

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    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 exponents v and 2 Δ4 -γ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation dv = 2 Δ4 -γ. In two dimensions, we confirm the predicted exponent v=3/4 and the hyperscaling relation; we estimate the universal ratios <Rg2>/<Re2>=0.14026±0.00007, <Rm2>/<Re2>=0.43961±0.00034, and Ψ*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimate v=0.5877±0.0006 with a correctionto-scaling exponent Δ1=0.56±0.03 (subjective 68% confidence limits). This value for v agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for Δ1. Earlier Monte Carlo estimates of v, which were ≈0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <Rg2>/<Re2>=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 relation dv = 2 Δ4 -γ for two-dimensional SAWs.

    Original languageEnglish (US)
    Pages (from-to)661-754
    Number of pages94
    JournalJournal of Statistical Physics
    Issue number3-4
    StatePublished - Aug 1 1995



    • Karp-Luby algorithm
    • Monte Carlo
    • Self-avoiding walk
    • critical exponent
    • hyperscaling
    • interpenetration ratio
    • pivot algorithm
    • polymer
    • renormalization group
    • second virial coefficient
    • two-parameter theory
    • universal amplitude ratio

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Mathematical Physics

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