Trapping and survival probability in two dimensions

We investigate the survival probability $\Phi \mathrm{}(n,c)\mathrm{}$ of particles performing a random walk on a two-dimensional lattice that contains static traps, which are randomly distributed with a concentration c, as a function of the number of steps n. $\Phi \mathrm{}(n,c)\mathrm{}$ is analyzed in terms of a scaling ansatz, which allows us to locate quantitatively the crossover between the Rosenstock approximation (valid only at early times) and the asymptotic Donsker-Varadhan behavior (valid only at long times). While the existence of the crossover has been postulated before, its exact location has not been known. Our scaling hypothesis is based on the mean value of the quantity $S_n,$ the number of sites visited in an n-step walk. We make use of the idea of self-interacting random walks, and a “slithering” snake algorithm, available in the literature, and we are thus able to obtain accurate survival probability data indirectly by Monte Carlo simulation techniques. The crossover can now be determined by our method, and it is found to depend on a combination of c and n. It occurs at small $\Phi \mathrm{}(n,c)\mathrm{}$ values, which is typically the case for large values of n.