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Dissipative Abelian sandpiles and random walks.

C Vanderzande1, F Daerden

  • 1Departement Wiskunde-Natuurkunde-Informatica, Limburgs Universitair Centrum, 3590 Diepenbeek, Belgium.

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

We found a connection between dissipative Abelian sandpiles and random walks. This reveals the correlation length exponent always equals 1/d(w), where d(w) is the random walker

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Area of Science:

  • Statistical Physics
  • Complex Systems

Background:

  • Dissipative Abelian sandpiles are complex systems exhibiting self-organized criticality.
  • Understanding their critical exponents is crucial for characterizing emergent behavior.

Purpose of the Study:

  • To establish a relationship between dissipative Abelian sandpiles and random walks.
  • To derive a general formula for the correlation length exponent (nu).

Main Methods:

  • Relating the sandpile model on graph L to a random walk on an extended graph with a trapping site.
  • Utilizing exact results and a scaling assumption.

Main Results:

  • Demonstrated that the correlation length exponent (nu) of dissipative sandpiles equals 1/d(w).
  • d(w) represents the fractal dimension of the random walker.
  • Confirmed the general result with exact data for finite Sierpinski gaskets.

Conclusions:

  • The study provides a new understanding of the correlation length exponent in dissipative sandpiles.
  • The derived relationship is applicable to various lattices, including Euclidean and fractal ones.
  • The findings offer a unified perspective on critical phenomena in sandpile models.

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