Critical Exponents, Hyperscaling and Universal Amplitude Ratios for Two- and Three-Dimensional Self-Avoiding Walks

Abstract

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponents $\nu$ and $2{\Delta }_4-\gamma$ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation $d\nu =2{\Delta }_4-\gamma$. In two dimensions, we confirm the predicted exponent $\nu =3/4$ and the hyperscaling relation; we estimate the universal ratios $\<R_g^2\> / \<R_e^2\> = 0.14026 \pm 0.00007$, $\<R_m^2\> / \<R_e^2\> = 0.43961 \pm 0.00034$ and ${\Psi }^{\ast }=0.66296\pm 0.00043$ (68\% confidence limits). In three dimensions, we estimate $\nu =0.5877\pm 0.0006$ with a correction-to-scaling exponent ${\Delta }_1=0.56\pm 0.03$ (subjective 68\% confidence limits). This value for $\nu$ agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for ${\Delta }_1$. Earlier Monte Carlo estimates of $\nu$, which were $\approx 0.592$, are now seen to be biased by corrections to scaling. We estimate the universal ratios $\<R_g^2\> / \<R_e^2\> = 0.1599 \pm 0.0002$ and ${\Psi }^{\ast }=0.2471\pm 0.0003$; since ${\Psi }^{\ast }>0$, hyperscaling holds. The approach to ${\Psi }^{\ast }$ 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.

87 pages including 12 figures, 1029558 bytes Postscript (NYU-TH-94/09/01)

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