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Origin of power-law distributions in deterministic walks: the influence of landscape geometry
M C Santos1, D Boyer, O Miramontes
1Departamento de Física, Universidade Federal do Paraná, 81531-990 Curitiba--PR, Brazil.
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.
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