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Random walks with self-similar clusters.

B D Hughes1, M F Shlesinger, E W Montroll

  • 1Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742.

Proceedings of the National Academy of Sciences of the United States of America
|June 1, 1981
PubMed
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This study introduces a lattice random walk model that mimics Lévy flights, revealing that infinite mean-squared displacement and transience are key for cluster formation. The walker effectively experiences a higher dimension due to fractal clusters.

Area of Science:

  • Statistical Physics
  • Complex Systems
  • Fractal Geometry

Background:

  • Random walks are fundamental models in statistical physics, used to describe diffusion and transport phenomena.
  • Lévy flights, a type of random walk with heavy-tailed jump distributions, exhibit anomalous diffusion and are relevant in various natural processes.
  • Understanding cluster formation in systems with complex jump dynamics remains a significant challenge.

Purpose of the Study:

  • To construct and analyze a discrete random walk model on a lattice with self-similar clusters.
  • To investigate the conditions necessary for cluster formation in such a system.
  • To explore the relationship between the random walk's dimensionality and the fractal dimension of the clusters.

Main Methods:

  • Construction of a random walk on a lattice where the jump distribution function incorporates a hierarchy of self-similar clusters.

Related Experiment Videos

  • Analysis of the random walk as a discrete analog of a Lévy flight, examining its continuum limit.
  • Mathematical analysis of the Fourier transform of the jump distribution, identifying it as a Weierstrass function.

    • Main Results:

    • Demonstration that an infinite mean-squared displacement per jump and transience of the random walk are necessary conditions for cluster formation.
    • The Fourier transform of the jump distribution function is identified as a continuous, nondifferentiable Weierstrass function.
    • The random walk exhibits an effective dimension higher than the spatial dimension, directly related to the fractal dimension of the self-similar clusters.

    Conclusions:

    • The developed random walk model provides a framework for understanding cluster formation driven by anomalous diffusion.
    • The findings highlight the critical role of jump characteristics (infinite mean-squared displacement, transience) in emergent cluster structures.
    • The concept of an effective dimensionality offers a novel perspective on how fractal geometry influences random walk behavior.