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Deterministic walks in random media.

G F Lima1, A S Martinez, O Kinouchi

  • 1Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Avenida Bandeirantes 3900, CEP 14040-901, Ribeirão Preto, SP, Brazil.

Physical Review Letters
|July 20, 2001
PubMed
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Deterministic walks explore nearest neighbor paths among random points. The study analyzes trajectory dynamics, finding transient times follow exponential laws and cycle densities follow power laws, with specific results for one-dimensional spaces.

Area of Science:

  • Complex Systems
  • Statistical Physics
  • Computational Geometry

Background:

  • Investigates deterministic walk algorithms on random point sets.
  • Considers a 'tourist' model visiting the nearest unvisited city within a 'tau' step memory.
  • Examines behavior in one and two dimensions (d=1, 2) with uniformly distributed points.

Purpose of the Study:

  • To characterize the trajectories generated by this nearest neighbor deterministic walk.
  • To analyze the statistical properties of transient times and cyclic attractors.
  • To derive analytical results for the one-dimensional case.

Main Methods:

  • Simulated deterministic walks on random point distributions.
  • Analysis of trajectory components: transient phase and p-cycle attractors.

Related Experiment Videos

  • Mathematical derivation of transient time distributions and cycle density functions.

    • Main Results:

    • Trajectories consist of a transient phase followed by a p-cycle attractor.
    • Transient times exhibit an exponential decay law dependent on the memory parameter 'tau'.
    • The density of p-cycles follows a power-law distribution D(p) ∝ p^(-alpha(tau)), with an exponent independent of 'tau' under specific conditions (τ>>1, τ/N<<1).

    Conclusions:

    • The deterministic nearest neighbor walk exhibits predictable statistical behavior.
    • The memory parameter 'tau' significantly influences transient dynamics but not the long-term cycle structure under certain limits.
    • Analytical insights were gained for the one-dimensional case, providing a foundation for higher dimensions.