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

This work investigates the properties of a deterministic walk, whose locomotion rule is always to travel to the nearest site, and finds scale invariance for an intermediate geometry, when the landscape is a thin striplike region.

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 (ALxL) landscape and, once reached, they become unavailable for future visits. As expected, the walker step lengths present characteristic scales in one (L-->0) and two (AL approximately L) dimensions. However, we find scale invariance for an intermediate geometry, when the landscape is a thin striplike 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.