MONTE-CARLO PRECISE DETERMINATION OF THE END-TO-END DISTRIBUTION FUNCTION OF SELF-AVOIDING WALKS ON THE SIMPLE-CUBIC LATTICE

Chains have been generated on the simple-cubic lattice to determine, by Monte Carlo simulation, the end-to-end distribution function of self-avoiding walks. The modulus r of the end-to-end distribution vector r, the square of this modulus, and the interactions of all orders were recorded for each chain. The Alexandrowicz dimerization procedure has been used to circumvent attrition and thus obtain statistically significant samples of large chains. This made it possible to obtain samples involving 12 000-16 000 chains, within "windows" of width DELTA-rho = 0.2, where rho = r/N-nu, N being the number of steps in the walk and nu the scaling exponent. It was found that the mean value <r> = aN-nu, with nu = 0.5919 and the prefactor a very close (perhaps strictly equal) to unity. The above value of nu is slightly larger than that calculated by Le Guillou and Zinn-Justin, but in accord with the Wilson epsilon = 4 - d expansion, where d is the space dimensionality. Also, <r2> = bN2-nu, with b = 1.136 +/- 0.05. An attempt was made to fit the data to the Fisher-McKenzie-des Cloizeaux distribution, P(r = N-nu-rho) = N-nu-d A rho-theta exp(-beta-rho-delta), where theta, beta, and delta are adjustable parameters. Taking delta = (1 - nu)-1 = 2.45, in accordance with the Fisher law, it was found that plots for various N values of ln L(rho) + beta-rho-delta vs ln-rho, where L(rho) is the number of chains falling within subwindows of width d-rho = 0.02, were linear (expect for the lowest values of rho), if beta was taken equal to 1.07. From the slope of the linear plot it was found that theta = 0.27, in accord with theory. Finally, in three dimensions the end-to-end distribution function P(r) can be expressed by the relation P(r = N-nu-rho) = N-nu-d 0.2393-rho-0.27 exp(-1.07-rho-2.45). This relation is in good agreement with that proposed by des Cloizeaux and Jannink. A brief discussion of the two-dimensional case as well as fluctuations of the end-to-end distance is also presented.