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Related Experiment Videos

Monochromatic path crossing exponents and graph connectivity in two-dimensional percolation.

Jesper Lykke Jacobsen1, Paul Zinn-Justin

  • 1Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France. jacobsen@lptms.u-psud.fr

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 7, 2003
PubMed
Summary

This study explores fractal dimensions in percolation clusters, revealing how the probability of traversing k disjoint paths changes with cluster connectivity. Results provide a new formula for predicting these probabilities in disordered systems.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Fractal Geometry

Background:

  • Percolation theory studies connectivity in random networks.
  • Fractal dimensions characterize the complex geometry of such networks.
  • Understanding k-connectivity is crucial for disordered systems.

Purpose of the Study:

  • To generalize fractal dimensions to k-connected parts of percolation clusters.
  • To investigate the asymptotic decay of probabilities for k disjoint paths.
  • To provide a numerical framework for k-connectivity analysis.

Main Methods:

  • Generalization of fractal dimensions for k-connectivity.
  • Analysis of probability decay using codimensions (x(k)).
  • Numerical computation via a transfer matrix approach for k up to 6.

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Main Results:

  • Numerical results for codimensions x(k) obtained for k=1 to 6.
  • A well-fitting ansatz derived for x(k): x(k) = 1/(12k^2) + 1/(48k) + (1-k)C.
  • Precise estimation of the constant C = 0.0181 ± 0.0006.

Conclusions:

  • The study successfully extends fractal dimension concepts to k-connected percolation clusters.
  • The derived ansatz accurately describes the behavior of k-disjoint path traversal probabilities.
  • Findings offer insights into the structure and connectivity of disordered systems.