SCALING EXPONENTS OF THE SELF-AVOIDING-WALK-PROBLEM IN 3 DIMENSIONS

Monte Carlo simulations on self-avoiding walks traced on the simple cu bic lattice and reported in a recent paper have been extended up to 29 99 steps, using the same Alexandrowicz dimerization procedure. Through this extension, we are able to show that the discrepancy between V-RG , the scaling exponent for the correlation length, as determined from renormalization-group calculations, and v(MC), the same exponent deter mined through Monte Carlo simulations, is an artefact, originating in the fact that Monte Carlo simulations are restricted to relatively sho rt chains, while to obtain the correct v value using the latter method exceedingly large chains are required. This finding is in accord with a previous suggestion by Zifferer. We further show that [r], the modu lus of the mean end-to-end distance, [r(2)](1/2), the root-mean-square end-to-end distance, [r(g)], the mean radius of gyration, and [r(g)(2 ) ](1/2), the root-mean-square radius of gyration, cannot be correctly expressed for all N in the range 1 < N < 2999 using a single correcti on to scaling exponent Delta(1). At least two such corrections to the scaling exponents are required, and the agreement with the Monte Carlo data is significantly improved if three corrections to the scaling ex ponents are introduced, so that one should write [x] = N-0.588[a(o)(x) + a(1)(x) N--Delta 1 + a(2)(x)N(-Delta 2) + a(3)(x)N(-Delta 3)], wher e [x] stands for one of the above mean values. Consideration of a four th correction to scaling exponent Delta(4) does not seem to be warrant ed for self-avoiding walks, where there is a lower cutoff for N = 1. F urther, such a fourth exponent seems devoid of physical significance b ecause of the even-odd oscillations occurring for the lowest N values in the various mean values [x], where a fourth exponent has a non-negl igible effect. A five-term expansion is however given here for complet eness. The set of the corrections to the scaling exponents Delta(i), w hich, because of universality, is the same for the various mean values , as well as of the a(i)(x')s, which depend on the mean value consider ed, follows the somewhat arbitrary choice made, within a narrow range of values, for the first correction to the scaling exponent Delta(1). If the value Delta(1) = 0.50 is adopted, as suggested by graphical ana lysis of our data, the set which minimizes the mean-square deviation o f the Monte Carlo data is Delta(2) = 1.0, Delta(3) = 2.0, and Delta(4) = 4.0. If the renormalization group value Delta(1) = 0.47 is used ins tead, the corresponding set is Delta(2) = 1.05+/-0.02, Delta(3) = 2.2/-0.2, and Delta(4) = 4.4+/-0.4. These two sets are mathematically equ ivalent for the correct description of our Monte Carlo data. The preci sion of our data does not permit one to decide which set, on a physica l basis, is the correct one, In any case, each successive correction t o the scaling exponent is found to be, approximately if not exactly, t he double of the preceding one.