ISSN:
1572-9613
Keywords:
Self-avoiding walk
;
polymer
;
critical exponent
;
hyperscaling
;
universal amplitude ratio
;
second virial coefficient
;
interpenetration ratio
;
renormalization group
;
two-parameter theory
;
Monte Carlo
;
pivot algorithm
;
Karp-Luby algorithm
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
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 g 2 〉/〈R e 2 〉=0.14026±0.00007, 〈R m 2 〉/〈R e 2 〉=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 g 2 〉/〈R e 2 〉=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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02178552
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