A myopic random walk on a finite chain

Abstract

We solve analytically the problem of a biased random walk on a finite chain of 'sites' (1,2,..,N) in discrete time, with 'myopic boundary conditions'-a walker at 1 (orN) at timen moves to 2 (orN - 1) with probability one at time (n + 1). The Markov chain has period two; there is no unique stationary distribution, and the moments of the displacement of the walker oscillate about certain mean values as n → ∞ , with amplitudes proportional to 1/N. In the continuous-time limit, the oscillating behaviour of the probability distribution disappears, but the stationary distribution is depleted at the terminal sites owing to the boundary conditions. In the limit of continuous space as well, the problem becomes identical to that of diffusion on a line segment with the standard reflecting boundary conditions. The first passage time problem is also solved, and the differences between the walks with myopic and reflecting boundaries are brought out.