Random walks on periodic and random lattices. II. Random walk properties via generating function techniques

Abstract

We investigate the random walk properties of a class of two-dimensional lattices with two different types of columns and discuss the dependence of the properties on the densities and detailed arrangements of the columns. We show that the row and column components of the mean square displacement are asymptotically independent of the details of the arrangement of columns. We reach the same conclusion for some other random walk properties (return to the origin and number of distinct sites visited) for various periodic arrangements of a given relative density of the two types of columns. We also derive exact asymptotic results for the occupation probabilities of the two types of distinct sites on our lattices which validate the basic conjecture on bond and step ratios made in the preceding paper in this series.