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

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 $\ / \ = 0.14026 \pm 0.00007$, $\ / \ = 0.43961 \pm 0.00034$ and $\Psi^* = 0.66296 \pm 0.00043$ (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 $\ / \ = 0.1599 \pm 0.0002$ and $\Psi^* = 0.2471 \pm 0.0003$; 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.