2000/05/25 by Gilson F. Lima, Lima, Gilson F., Alexandre Souto Martinêz +3
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.cond-mat/0005446
openalex publication_date 2000/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Deterministic walks over a random set of points in one and two dimensions (d=1,2) are considered. Points (``cities'') are randomly scattered in Rd following a uniform distribution. A walker (a ``tourist''), at each time step, goes to the nearest neighbor city that has not been visited in the past τsteps. Each initial city leads to a trajectory composed of a transient part and a final p-cycle attractor. The distribution of transient times, p-cycles and number of cities per attractor are studied. It is shown numerically that transient times (for d=1,2) follow a Poisson law with a τ dependent decay but the density of p-cycles follows a power law D(p) ∝ p-α(τ) for d=2. For large τ, the expoent tends to α~ 5/2. Some analytical results are given for the d=1 case. Since the power law is robust and does not depend on free parameters, this system presents ``generic scale invariance''. Applications to animal exploratory behavior and other local minimization problems are suggested.