The Pearson walk with shrinking steps in two dimensions

We study the shrinking Pearson random walk in two dimensions and greater, in which the direction of the Nth step is random and its length equals λN−1, with λ<1. As λ increases past a critical value λc, the endpoint distribution in two dimensions, P(r), changes from having a global maximum away from the origin to being peaked at the origin. The probability distribution for a single coordinate, P(x), undergoes a similar transition, but exhibits multiple maxima on a fine length scale for λ close to λc. We numerically determine P(r) and P(x) by applying a known algorithm that accurately inverts the exact Bessel function product form of the Fourier transform for the probability distributions