Monte Carlo precise determination of the end-to-end distribution function of self-avoiding walks on the simple-cubic lattice

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Abstract
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 Δρ=0.2, where ρ=r/Nν, N being the number of steps in the walk and ν the scaling exponent. It was found that the mean value 〈r〉=aNν, with ν=0.5919 and the prefactor a very close (perhaps strictly equal) to unity. The above value of ν is slightly larger than that calculated by Le Guillou and Zinn-Justin, but in accord with the Wilson ε=4−d expansion, where d is the space dimensionality. Also, 〈r 2〉=bN2ν, with b=1.136±0.05. An attempt was made to fit the data to the Fisher–McKenzie–des Cloizeaux distribution, P(r=Nνρ)=N−νdA ρθ exp(−βρδ), where θ, β, and δ are adjustable parameters. Taking δ=(1−ν)−1=2.45, in accordance with the Fisher law, it was found that plots for various N values of ln L(ρ)+βρδ vs ln ρ, where L(ρ) is the number of chains falling within subwindows of width dρ=0.02, were linear (except for the lowest values of ρ), if β was taken equal to 1.07. From the slope of the linear plot it was found that θ=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=Nνρ)=N−νd 0.2393ρ0.27 exp(−1.07ρ2.45). 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.