Persistent 2D random walk with geometrically shrinking steps with four directions

Abstract

In this paper we present the steady state approximation of the probability density plots of a random walk with geometrically shrinking steps with persistence that may only take steps in four orthogonal directions on a 2D plane. Unlike its isotropic counterpart in which theprobability density appear continuous after N = 5 steps, the probability density of the random walk considered remains discontinuous even after N = 8 steps, a feature that is retained from its 1D counterpart. The shrinking steps, attributed to the shrinking ratio λ < 1, makes the random walk a bounded diffusion. Associated to the four possible step directions, the random walker is confined within a square of dimensions 2λ/(1 − λ) × 2λ/(1 −λ). The effect of the persistence factor either gathers the probability density near the center or near the vertices of the square for α < 0 and α > 0 respectively.