Scanning method as an unbiased simulation technique and its application to the study of self-attracting random walks.

Abstract

The scanning method proposed by us [J. Phys. A 15, L735 (1982); Macromolecules 18, 563 (1985)] for simulation of polymer chains is further developed and applied, for the first time, to a model with finite interactions. In addition to ``importance sampling,'' we remove the bias introduced by the scanning method with a procedure suggested recently by Schmidt [Phys. Rev. Lett. 51, 2175 (1983)]; this procedure has the advantage of enabling one to estimate the statistical error. We find these two procedures to be equally efficient. The model studied is an N-step random walk on a lattice, in which a random walk i has a statistical weight , where p1 is an attractive energy parameter and ${M}_{i}$ is the number of distinct sites visited by walk i.This model, which corresponds to a model of random walks moving in a medium with randomly distributed static traps, has been solved analytically for N\ensuremath{\rightarrow}\ensuremath{\infty} for any dimension d by Donsker and Varadhan (DV) and by others. 〈M〉 and ln\ensuremath{\varphi}, where \ensuremath{\varphi} is the survival probability in the trapping problem, diverge like ${N}^{\ensuremath{\alpha}}$ with \ensuremath{\alpha}=d/(d+2). Most numerical studies, however, have failed to reach the DV regime in which d/(d+2) becomes a good approximation for \ensuremath{\alpha}. On the other hand, our results for \ensuremath{\alpha} (obtained for N\ensuremath{\le}150) are close to the DV values for p\ensuremath{\le}0.7 and p\ensuremath{\le}0.6 for d=2 and 3, respectively.This suggests that the scanning method is more efficient than both the commonly used direct Monte Carlo technique, and the Rosenbluth and Rosenbluth method [J. Chem. Phys. 23, 356 (1954)]. Our results support the conclusion of Havlin et al. [Phys. Rev. Lett. 53, 407 (1984)] that the DV regime exists already for \ensuremath{\varphi}\ensuremath{\le}${10}^{\mathrm{\ensuremath{-}}13}$ for both d=2 and 3. We also find that at the percolation threshold ${p}_{c}$ the exponents for the end-to-end distance are small, but larger than zero, and that the probability of a walk returning to the origin behaves approximately as ${N}^{\mathrm{\ensuremath{-}}1/3}$ for both d=2 and 3.