One-Dimensional Random Walk in Alternating Homogeneous Domains

We consider the prototypical case of a lattice that is homogeneous in the large but inhomogeneous in the small-a one-dimensional random walk with alternating homogeneous lattice fragments and appropriate boundary probabilities. The generating function for the probability distribution of being at position x after N steps is obtained. We also find various asymptotic forms and limiting distributions as a function of the step probability and the lattice fragments.