Scaling exponents of the self-avoiding-walk-problem in three dimensions.

Abstract

Monte Carlo simulations on self-avoiding walks traced on the simple cubic lattice and reported in a recent paper have been extended up to 2999 steps, using the same Alexandrowicz dimerization procedure. Through this extension, we are able to show that the discrepancy between ${\ensuremath{\nu}}_{\mathrm{RG}}$, the scaling exponent for the correlation length, as determined from renormalization-group calculations, and ${\ensuremath{\nu}}_{\mathrm{MC}}$, the same exponent determined through Monte Carlo simulations, is an artefact, originating in the fact that Monte Carlo simulations are restricted to relatively short chains, while to obtain the correct \ensuremath{\nu} 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 modulus of the mean end-to-end distance, 〈${\mathit{r}}^{2}$${\mathrm{〉}}^{1/2}$, the root-mean-square end-to-end distance, 〈${\mathit{r}}_{\mathit{g}}$〉, the mean radius of gyration, and 〈${\mathit{r}}_{\mathit{g}}^{2}$${\mathrm{〉}}^{1/2}$, the root-mean-square radius of gyration, cannot be correctly expressed for all N in the range 1N2999 using a single correction to scaling exponent ${\mathrm{\ensuremath{\Delta}}}_{1}$. 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 exponents are introduced, so that one should write 〈x〉=${\mathit{N}}^{0.588}$[${\mathit{a}}_{0}^{\mathit{x}}$+${\mathit{a}}_{1}^{\mathit{x}}$${\mathit{N}}_{1}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}}$ +${\mathit{a}}_{2}^{\mathit{x}}$${\mathit{N}}_{2}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}}$+${\mathit{a}}_{3}^{\mathit{x}}$${\mathit{N}}_{3}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}}$], where 〈x〉 stands for one of the above mean values.Consideration of a fourth correction to scaling exponent ${\mathrm{\ensuremath{\Delta}}}_{4}$ does not seem to be warranted for self-avoiding walks, where there is a lower cutoff for N=1. Further, such a fourth exponent seems devoid of physical significance because of the even-odd oscillations occurring for the lowest N values in the various mean values 〈x〉, where a fourth exponent has a non-negligible effect. A five-term expansion is however given here for completeness. The set of the corrections to the scaling exponents ${\mathrm{\ensuremath{\Delta}}}_{\mathit{i}}$, which, because of universality, is the same for the various mean values, as well as of the ${\mathit{a}}_{\mathit{i}}^{\mathit{x}}$'s, which depend on the mean value considered, follows the somewhat arbitrary choice made, within a narrow range of values, for the first correction to the scaling exponent ${\mathrm{\ensuremath{\Delta}}}_{1}$. If the value ${\mathrm{\ensuremath{\Delta}}}_{1}$=0.50 is adopted, as suggested by graphical analysis of our data, the set which minimizes the mean-square deviation of the Monte Carlo data is ${\mathrm{\ensuremath{\Delta}}}_{2}$=1.0, ${\mathrm{\ensuremath{\Delta}}}_{3}$=2.0, and ${\mathrm{\ensuremath{\Delta}}}_{4}$=4.0. If the renormalization group value ${\mathrm{\ensuremath{\Delta}}}_{1}$=0.47 is used instead, the corresponding set is ${\mathrm{\ensuremath{\Delta}}}_{2}$=1.05\ifmmode\pm\else\textpm\fi{}0.02, ${\mathrm{\ensuremath{\Delta}}}_{3}$=2.2\ifmmode\pm\else\textpm\fi{}0.2, and ${\mathrm{\ensuremath{\Delta}}}_{4}$=4.4\ifmmode\pm\else\textpm\fi{}0.4. These two sets are mathematically equivalent for the correct description of our Monte Carlo data. The precision of our data does not permit one to decide which set, on a physical basis, is the correct one. In any case, each successive correction to the scaling exponent is found to be, approximately if not exactly, the double of the preceding one.