Exact-enumeration approach to random walks on percolation clusters in two dimensions

We present a useful method for the study of random walks on disordered systems, and apply it to the problem of diffusion on percolation clusters at criticality. The method is based on the exact enumeration of all possible random walks of a certain size on a given cluster. In particular, we calculate the meansquare end-to-end distance, the probability of return to the origin, and a diffusion chemical exponent $d_w^l$ (that describes the chemical distance traveled by the random walker) as functions of the number of steps. Also we present for the first time data showing clearly the difference between the myopic and blind ants, and find much more rapid convergence for the blind ant.