Deterministic walks in random media

Deterministic walks over a random set of N points in one and two dimensions (d = 1,2) are considered. Points ("cities") are randomly scattered in 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) follow an exponential law with a tau -dependent decay time but the density of p cycles can be approximately described by D(p) proportional to p(-alpha(tau)). For tau >> 1 and tau /N << 1, the exponent is independent of tau. Some analytical results are given for the d = 1 case.