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Simple model of a random walk with arbitrarily long memory.

A Berrones1, H Larralde

  • 1Centro de Ciencias Físicas, Universidad Nacional Autónoma de México, Facultad de Ciencias, UAEMo, Cuernavaca, Morelos, Mexico.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 20, 2001
PubMed
Summary

We introduce a generalized persistent random-walk model with long memory effects. This model remains mathematically tractable, allowing analysis of its statistical properties.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Mathematical Modeling

Background:

  • Persistent random walks are fundamental models in statistical mechanics.
  • Standard models often assume short or no memory between steps.
  • Introducing memory effects can capture complex system dynamics.

Purpose of the Study:

  • To generalize the persistent random-walk model to include long-range memory.
  • To demonstrate the mathematical tractability of the generalized model.
  • To analyze the statistical properties of delayed step persistence.

Main Methods:

  • Generalization of the persistent random-walk model with a delay parameter T.
  • Development of a transformation to maintain model tractability.
  • Asymptotic statistical analysis of the delayed persistence case.

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Main Results:

  • The generalized model exhibits arbitrarily long memory effects.
  • The model is shown to be tractable using established techniques.
  • Asymptotic properties for delayed step persistence were derived.

Conclusions:

  • The generalized persistent random-walk model effectively incorporates long memory.
  • The model provides a tractable framework for studying complex systems.
  • This work extends the applicability of random-walk models to systems with delayed correlations.