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1.
Let
l
be the critical exponent associated with the probability thatl independentN-step ordinary random walks, starting at nearby points, are mutually avoiding. Using Monte Carlo methods combined with a maximum-likelihood data analysis, we find that in two dimensions 2=0.6240±0.0005±0.0011 and 3=1.4575±0.0030±0.0052, where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second error bar represents statistical error (classical 95% confidence limits). These results are in good agreement with the conformal-invariance predictions 2=5/8 and 3=35/24. 相似文献
2.
3.
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 2
4 – as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2
4 –. 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 *=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponent
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 for
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 *=0.2471±0.0003; since *>0, hyperscaling holds. The approach to * 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 = 2
4 – for two-dimensional SAWs. 相似文献
4.
5.
We introduce a new Monte Carlo algorithm for the self-avoiding walk (SAW), and show that it is particularly efficient in the critical region (long chains). We also introduce new and more efficient statistical techniques. We employ these methods to extract numerical estimates for the critical parameters of the SAW on the square lattice. We find=2.63820 ± 0.00004 ± 0.00030=1.352 ± 0.006 ± 0.025v=0.7590 ± 0.0062 ± 0.0042 where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second bar represents statistical error (classical 95% confidence limits). These results are based on SAWs of average length 166, using 340 hours CPU time on a CDC Cyber 170–730. We compare our results to previous work and indicate some directions for future research. 相似文献
6.
7.
Alan D. Sokal 《Probability Theory and Related Fields》1981,56(4):537-548
Summary Let (, , ) be a perfect probability space with countably generated, and let IB be a family of sub--fields of . Under a countability condition on the family IB, I show that there exists a family {}IB of regular conditional probabilities which are everywhere compatible. Under a more stringent condition on IB, I show that the
can furthermore be chosen to be everywhere proper. It follows that in the Dobrushin-Lanford-Ruelle formulation of the statistical mechanics of classical lattice systems, every (perfect) probability measure is a Gibbs measure for some specification.Research supported in part by NSF PHY-78-23952NSF Predoctoral Fellow (1976–79) and Danforth Fellow (1979–81). 相似文献
8.
We study the antiferromagnetic q-state Potts model on the square lattice for q=3 and q=4, using the Wang–Swendsen–Kotecký (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q=3 we obtain good control up to correlation length 5000; the data are consistent with ()=Ae
2
p
(1+a
1
e
–
+ ...) as , with p1. The staggered susceptibility behaves as
stagg
5/3. For q=4 the model is disordered (2) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice. 相似文献
9.
We prove that theq-state Potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) wheneverq>2r. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay forq7), triangular lattice (q11), hexagonal lattice (q4), and Kagomé lattice (q6). The proofs are based on the Dobrushin uniqueness theorem. 相似文献
10.
At zero temperature, the 3-state antiferromagnetic Potts model on a square lattice maps exactly onto a point of the 6-vertex model whose long-distance behavior is equivalent to that of a free scalar boson. We point out that at nonzero temperature there are two distinct types of excitation: vortices, which are relevant with renormalization-group eigenvalue 1/2 and non-vortex unsatisfied bonds, which are strictly marginal and serve only to renormalize the stiffness coefficient of the underlying free boson. Together these excitations lead to an unusual form for the corrections to scaling: for example, the correlation length diverges as J/kT according to Ae
2
(1+be
–
+···), where b is a nonuniversal constant that may nevertheless be determined independently. A similar result holds for the staggered susceptibility. These results are shown to be consistent with the anomalous behavior found in the Monte Carlo simulations of Ferreira and Sokal. 相似文献