Random Walks on Quasi-One-Dimensional Lattices

We use the well-known generating function technique to study the asymptotic behavior of basic statistical properties for the separable continuous-time random walks on quasi-one-dimensional lattices, consisting of periodically repeated unit cells. In each unit cell, there exists a particular site named a major site, which is connected to its equivalent sites in neighboring unit cells by a certain kind of network. In addition, there may be a dangling network attached to the major site. Relying on the existence of such sites and using the generating function technique, we obtain exact analytic expressions of the first and the second moments of the walker location along the structure axis and other random walk properties for a given major site in the form of the probability distribution function of waiting time and the multi-step transition probabilities. As an application, we study random walks on sparse comb structures and show the simulation results which perfectly support our analytical results.