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 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponents v and 2 Δ₄ -γ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation dv = 2 Δ₄ -γ. In two dimensions, we confirm the predicted exponent v=3/4 and the hyperscaling relation; we estimate the universal ratios <R_g²>/<R_e²>=0.14026±0.00007, <R_m²>/<R_e²>=0.43961±0.00034, and Ψ^*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimate v=0.5877±0.0006 with a correctionto-scaling exponent Δ₁=0.56±0.03 (subjective 68% confidence limits). This value for v agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for Δ₁. Earlier Monte Carlo estimates of v, which were ≈0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R_g²>/<R_e²>=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 relation dv = 2 Δ₄ -γ for two-dimensional SAWs.