Exact distribution function for discrete time correlated random walks in one dimension

Abstract

A discrete time correlated random walk in one dimension is investigated. Combinatorial arguments are used to calculate the exact probability distribution PN(L), the probability that the correlated random walker arrives at a distance L steps to the right of its starting point after exactly N steps. PN(L) is calculated using arbitrary initial conditions which permit the influence of end effects and boundary conditions to be calculated and several special cases are considered in detail. PN(L) with arbitrary initial conditions is calculated both with and without a bias for motion in one direction yielding a useful model for the combined diffusion and drift of charged particles undergoing a correlated random walk in an applied field. The relation of the correlated random walk to the Ising model is also discussed. © 1998 American Institute of Physics.