CRITICAL EXPONENTS, HYPERSCALING, AND UNIVERSAL AMPLITUDE RATIOS FOR 2-DIMENSIONAL AND 3-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 nu and 2 Delta(4) - y as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation dv = 2 Delta(4) - y. In two dimensions, we confirm the predicted exponent v=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 Psi* = 0.66296 +/- 0.00043 (68% confidence limits). In three dimensions, we estimate v = 0.5877 +/- 0.0006 with a correction-to-scaling exponent Delta(1)=0.56 +/- 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 v, which were approximate to 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 Psi* = 0.2471 +/- 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. In an appendix, we prove rigorously (module some standard scaling assumptions)the hyperscaling relation dv = 2 Delta(4) - y for two-dimensional SAWs.