Correlated random-walk simulations in simple and binary lattices

Abstract

Random-walk simulations were performed on one-, two-, and three-dimensional, simple and binary lattices with several coordination numbers containing about one million sites. The random walk included a correlation parameter $l$ (Gaussian distribution with given standard deviation) representing a partial directional memory. The walks on the random binary lattices were constrained to sites of one component only (concentration $C$) with the sites of the second component acting as reflecting microboundaries. All simulations were restricted to the percolating cluster. The simple lattice simulations are compared with the well-known asymptotic analytical expressions for simple random walk ($l=1$) and with an expression for correlated walks ($l\ensuremath{\gg}1$). The visitation efficiency increases, as expected, with $C$. It also increases with $l$ for simple and high-$C$ lattices. However, for lower-$C$ lattices the visitation efficiency decreases with $l$, thus giving rise to "crossover concentrations." Our results are given in a series of figures of the efficiency or the number of sites visited versus the number of steps, showing the effects of concentration ($C$), and correlation ($l$). Applications to exciton percolation and coherence are mentioned.