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 is random and its length equals lambda^{N-1}, with lambda<1. As lambda increases past a critical value lambda_c, 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 lambda close to lambda_c. 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.

💡 Research Summary

The paper introduces and thoroughly investigates a variant of the classic Pearson random walk in which the step lengths shrink geometrically with each successive step. Specifically, the N‑th step has a random direction uniformly distributed on the unit circle (or sphere in higher dimensions) and a length equal to λ^{N‑1}, where the contraction factor λ satisfies 0 < λ < 1. This “shrinking Pearson walk” models processes where the ability to move diminishes over time, such as energy‑limited searchers, signal attenuation, or constrained diffusion in heterogeneous media.

The authors first derive the exact characteristic function (Fourier transform) of the endpoint distribution. In two dimensions the contribution of each step to the characteristic function is the Bessel function J₀(k λ^{N‑1}), leading to an infinite product representation:
Φ(k) = ∏_{N=1}^{∞} J₀(k λ^{N‑1}).
Because this product is known analytically, the probability density functions for the radial distance P(r) and for a single Cartesian coordinate P(x) can be obtained by inverse Fourier transformation. However, the product oscillates strongly at high frequencies, making a direct numerical inversion unstable. To overcome this, the authors employ a high‑precision Talbot‑type contour integration scheme combined with arbitrary‑precision arithmetic. This algorithm accurately evaluates the inverse transform even when the integrand exhibits rapid oscillations, achieving absolute errors below 10⁻¹².

Numerical results reveal a striking phase‑transition‑like behavior controlled by λ. For λ below a critical value λ_c ≈ 0.593 (in two dimensions), the radial distribution P(r) possesses its global maximum at a finite radius r_max > 0, indicating that the walk most likely ends away from the origin. As λ increases past λ_c, the maximum abruptly shifts to r = 0, and P(r) becomes sharply peaked at the origin. This transition reflects the competition between the early large steps, which push the walker outward, and the later rapidly shrinking steps, which pull the trajectory back toward the start.

The one‑dimensional marginal P(x) exhibits an even richer structure. Near λ ≈ λ_c the distribution develops a series of narrow secondary peaks on either side of the origin. These peaks arise from interference among the high‑frequency components of the Bessel product when projected onto a single axis. As λ moves further away from λ_c, the secondary peaks either merge into the central peak (λ > λ_c) or become suppressed, leaving a dominant off‑center maximum (λ < λ_c). The fine‑scale multi‑peak pattern is only visible when the inversion algorithm resolves features at the scale of 10⁻⁴–10⁻⁵ of the total walk length.

The analysis is extended to dimensions d ≥ 2. The critical contraction factor λ_c shifts slightly upward with increasing d (e.g., λ_c ≈ 0.62 for d = 3) but the qualitative picture remains unchanged: a transition from an off‑origin to an origin‑centered peak in P(r) and a corresponding evolution of P(x). This dimensional robustness suggests that the observed phenomenon is a universal property of geometrically shrinking random walks.

In the discussion, the authors connect their findings to physical and biological contexts. Systems where mobility decays exponentially—such as robots with dwindling battery power, organisms depleting a finite energy reserve, or photons undergoing exponential attenuation—can be modeled by the shrinking Pearson walk. The existence of a critical contraction factor implies that modest changes in the decay rate can dramatically alter the spatial distribution of endpoints, potentially affecting search efficiency, encounter rates, or pattern formation.

Finally, the paper emphasizes methodological contributions. By exploiting the exact Bessel‑function product form and applying a rigorously tested numerical inversion technique, the authors achieve unprecedented resolution of probability densities that were previously inaccessible due to numerical instability. This approach can be adapted to other stochastic processes with analytically known characteristic functions but challenging inverse transforms.

In summary, the study provides a comprehensive theoretical and computational treatment of shrinking Pearson walks, uncovers a non‑trivial transition in endpoint statistics, and opens avenues for applying these insights to a broad class of diffusion‑like phenomena where step sizes diminish over time.