Random Walk with a Hop-Over Site: A Novel Approach to Tagged Diffusion and Its Applications

We study, on a $d$ dimensional hypercubic lattice, a random walk which is homogeneous except for one site. Instead of visiting this site, the walker hops over it with arbitrary rates. The probability distribution of this walk and the statistics associated with the hop-overs are found exactly. This analysis provides a simple approach to the problem of tagged diffusion, i.e., the movements of a tracer particle due to the diffusion of a vacancy. Applications to vacancy mediated disordering are given through two examples.