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 $(A∕L\times L)$ 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 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.

Figure 1

Schematics of the search space where the small squares represent the randomly distributed target sites.

Figure 2

${\ell }_0$ as function of $L∕{\ell }_0$. For $L∕{\ell }_0⪢1$ (2D regime), ${\ell }_0$ is the constant $1∕\sqrt{N}$, whereas for $L∕{\ell }_0⪡1$ (1D), ${\ell }_0$ goes as $1∕\left(NL\right)$. The crossover takes place for $L∕{\ell }_0$ around unity.

Figure 3

The numerically calculated distributions of distances between closest neighbor sites for three values of $L∕{\ell }_0$ (see main text), corresponding to the 2D (cross), 1D (triangle), and crossover region (diamond) cases. The respective analytical distributions fitting 2D and 1D are plotted as solid lines. Note that the intermediate (crossover) is close to the 2D case.

Figure 4

Normalized step length distributions for the same parameters as in Fig. 3. (a) The triangles (cross) represents the 1D (2D) limit. The curves show the fits $1.3\exp [-0.92\ell ∕{\ell }_0]$ (dashed) and $3.1{(\ell ∕{\ell }_0)}^{-5.3}$ (continuous). (b) The intermediate case (diamond). Here the fit is $0.23{(\ell ∕{\ell }_0)}^{-2.2}$.

Figure 5

(a) The numerical average step length $\overline{\ell }$ (in units of ${\ell }_0$) taken by the walker during the search as function of $L∕{\ell }_0$. The region where $\overline{\ell }$ presents a peak, indicated by a rectangle, is shown in detail in (b). The continuous curves are just guides for the eye.

Figure 6

The distribution $P(\ell ∕{\ell }_0)$ fitted as ${(\ell ∕{\ell }_0)}^{-\mu }$. The parameters are $L∕{\ell }_0=2.37957$ and $\mu =2.3155$ (open triangle); $L∕{\ell }_0=4.21598$ and $\mu =2.22267$ (full circle); $L∕{\ell }_0=9.74390$ and $\mu =2.40229$ (open circle); $L∕{\ell }_0=22.1615$ and $\mu =2.65706$ (full square); and $L∕{\ell }_0=38.2288$ and $\mu =2.96385$ (open square).

Figure 7

Angular distribution of the turning angles between consecutive steps. The triangles (1D), crosses (2D), and diamonds (crossover region) correspond to the same cases of Fig. 4. The dotted curve (for the triangles) is just a guide for the eye.

Figure 8

In the crossover regime, the search space is a narrow strip. In such case, a walker moving in average to the right may perform many steps in the opposite direction, returning near the rightmost visited site through a very long step.

Figure 9

Number of steps vs. the horizontal projection of the walker position, in the 2D (a), crossover region (b), and 1D (c) cases of Fig. 4. Notice the crucial ultralong steps at the crossover geometry (b).

Figure 10

(a) The numerical drift velocity along $x$, $⟨\mid x-x_0\mid ∕n⟩∕{\ell }_0$, as a function of $L∕{\ell }_0$. The rectangle indicates part of the crossover region where there is an inflection point for the drift. (b) Blow up of the marked region in (a), showing a local minimum around $L∕{\ell }_0=1.7094$. The continuous curves are just guides for the eye.

Figure 11

The numerical fraction of visited targets, $\chi$, as a function of $L∕{\ell }_0$. The inset shows a local minimum in the crossover region. The continuous curves are just guides for the eye.