Exact and efficient discrete random walk method for time-dependent two-dimensional environments

We present an exact method for speeding up random walk in two-dimensional complicated lattice environments. To this end, we derive the discrete two-dimensional probability distribution function for a diffusing particle starting at the center of a square of linear size s. This is used to propagate random walkers from the center of the square to sites which are nearest neighbors to its perimeter sites, thus saving O(s(2)) steps in numerical simulations. We discuss in detail how this method can be implemented efficiently. We examine its performance in the diffusion limited aggregation model which produces fractal structures, and in a one-sided step-growth model producing compact, fingerlike structures. We show that in both cases, the square propagator method reduces the computational effort by a factor proportional to the linear system size as compared to standard random walk.