First-passage probabilities and mean number of sites visited by a persistent random walker in one- and two-dimensional lattices.

Abstract

We study the first passage probability and mean number of sites visited by a discrete persistent random walker on a lattice in one and two dimensions. This is performed by using the multistate formulation of the process. We obtain explicit expressions for the generating functions of these quantities. To evaluate these expressions, we study the system in the strongly persistent limit. In the one-dimensional case, we recover the behavior of the continuous one-dimensional persistent random walk (telegrapher process). In two dimensions we obtain an explicit expression for the probability distribution in the strongly persistent limit, however, the Laplace transforms required to evaluate the first-passage processes could only be evaluated in the asymptotic limit corresponding to long times in which regime we recover the behavior of normal two-dimensional diffusion.