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 τ 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 τ-dependent decay time but the density of p cycles can be approximately described by D(p)∝p^-α(τ). For τ≫1 and τ/N≪1, the exponent is independent of τ. Some analytical results are given for the d = 1 case.