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

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 ($A\mathrm{/}L\times L)$ 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.