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-dimension al self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exp onents nu and 2 Delta(4) - y as well as several universal amplitude ra tios; in particular, we make an extremely sensitive test of the hypers caling relation dv = 2 Delta(4) - y. In two dimensions, we confirm the predicted exponent v=3/4 and the hyperscaling relation; we estimate t he 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 (s ubjective 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 correcti ons 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, hypers caling holds. The approach to Psi is from above, contrary to the pred iction of the two-parameter renormalization-group theory. We criticall y reexamine this theory, and explain where the error lies. In an appen dix, we prove rigorously (module some standard scaling assumptions)the hyperscaling relation dv = 2 Delta(4) - y for two-dimensional SAWs.