Critical exponents, hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks
- World Financial Center, New York, NY (United States)
- York Univ., North York, Ontario (Canada)
- New York Univ., NY (United States)
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{triangle}{sub 4}-{gamma} as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation dv=2{triangle}{sub 4}-{gamma}. In two dimensions, we confirm the predicted exponent v=3/4 and the hyperscaling relation; we estimate the universal ratios / = 0.14026{+-}0.00007, / = 0.43961{+-}0.00034, and {Psi}* = 0.66296{+-}0.00043 (68% confidence limits). In three dimensions, we estimate v = 0.5877{+-}0.0006 with a correction-to-scaling exponent {triangle}{sub 1} = 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 {triangle}1. Earlier Monte Carlo estimates of v, which were {approximately}0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios / = 0.1599{+-}0.0002 and {Psi}* = 0.2471{+-}0.0003; since {Psi}* >0, hyperscaling holds. The approach to {Psi}* 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{triangle}{sub 4}-{gamma} for two-dimensional SAWs.
- Sponsoring Organization:
- USDOE
- OSTI ID:
- 217887
- Journal Information:
- Journal of Statistical Physics, Vol. 80, Issue 3-4; Other Information: PBD: Aug 1995
- Country of Publication:
- United States
- Language:
- English
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