Deterministic walks over a random set of $N$ points in one and two dimensions ( $d=1,2\mathrm{}$) are considered. Points (“cities”) are randomly scattered in ${\mathbf{R}}^d$ following a uniform distribution. A walker (“tourist”), at each time step, goes to the nearest neighbor city that has not been visited in the past $\tau$ steps. Each initial city leads to a different trajectory composed of a transient part and a final $p$-cycle attractor. Transient times (for $d=1,2\mathrm{}$) follow an exponential law with a $\tau$-dependent decay time but the density of $p$ cycles can be approximately described by $D\left(p\right)\propto p^{-\alpha \left(\tau \right)}$. For $\tau \gg 1$ and $\tau /N\ll 1$, the exponent is independent of $\tau$. Some analytical results are given for the $d=1\mathrm{}$ case.

• Received 5 May 2000

DOI:https://doi.org/10.1103/PhysRevLett.87.010603