Diffusion with Random Distribution of Static Traps

The survival probability $P(c,t)$ of a random walk of $t$ steps with static traps at concentration $c$ is studied in two and three dimensions by an efficient Monte Carlo method based on a mapping onto a polymer model. On the basis of the theoretical work of Donsker and Varadhan [Commun. Pure Appl. Math. 28, 525 (1975); 32, 721 (1979)] and of Rosenstock [J. Math. Phys. (N.Y.) 11, 487 (1970)] one expects a data collapse for $-\ln \left[P\right(c,t\left)\right]/\ln \left(t\right)$ plotted vs $\sqrt{\lambda t}/\ln \left(t\right)$ [with $\lambda =-\ln (1-c)$], in two dimensions, and for ${-t}^{-1/3\mathrm{}}\ln \left[P\right(c,t\left)\right]$ vs $t^{2/3}\lambda$ in three dimensions. These predictions are well supported by the Monte Carlo results.