The origin of power-law distributions in deterministic walks: the influence of landscape geometry

TL;DR

This paper explores how the geometry of a landscape influences the step length distribution in a deterministic walk, revealing a power-law distribution in a specific intermediate geometry due to a trapping mechanism.

Contribution

It demonstrates that landscape geometry can induce scale invariance and power-law distributions in deterministic walks, a novel insight into spatially constrained movement models.

Findings

Power-law distribution of step lengths in a thin strip landscape

Characteristic scales in one and two-dimensional landscapes

Dynamical trapping mechanism causes scale invariance

Abstract

We investigate the properties of a deterministic walk, whose locomotion rule is always to travel to the nearest site. Initially the sites are randomly distributed in a closed rectangular ($A/L \times L)$ landscape and, once reached, they become unavailable for future visits. As expected, the walker step lengths present characteristic scales in one ($L \to 0$) and two ($A/L \sim L$) dimensions. However, we find scale invariance for an intermediate geometry, when the landscape is a thin strip-like region. This result is induced geometrically by a dynamical trapping mechanism, leading to a power law distribution for the step lengths. The relevance of our findings in broader contexts -- of both deterministic and random walks -- is also briefly discussed.