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 $2{\mathrm{\Delta }}_4-\gamma$ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation $d\nu =2{\mathrm{\Delta }}_4-\gamma$. In two dimensions, we confirm the predicted exponent $\nu =3\mathrm{/}4$ and the hyperscaling relation; we estimate the universal ratios $\text{<}{R}_g^2\mathrm{/}\text{<}{R}_e^2=0.14026\pm 0.00007$, $<R_m^2> / <R_e^2> = 0.43961 \pm 0.00034$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 ${\mathrm{\Delta }}_1=0.56\pm 0.03$ (subjective 68% confidence limits). This value for $\nu$ agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for ${\mathrm{\Delta }}_1$. 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 $\text{<}{R}_g^2\mathrm{/}\text{<}{R}_e^2=0.1599\pm 0.0002$ and $\Psi^* = 0.2471 \pm 0.0003$;since$\Psi^* > 0$,hyperscalingholds\mathrm{.}Theapproachto$\Psi^*$ 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.