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Related Experiment Videos

Scaling for the critical percolation backbone.

M Barthélémy1, S V Buldyrev, S Havlin

  • 1Center for Polymer Studies and Deptartment of Physics, Boston University, Boston, Massachusetts 02215, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
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We analyzed the backbone mass in a 2D lattice at the percolation threshold. The average backbone mass scales with system size and distance, revealing critical exponents for lattice structures.

Area of Science:

  • Statistical physics
  • Complex systems
  • Network science

Background:

  • Percolation theory describes the formation of connected clusters in random systems.
  • Understanding the properties of the backbone, the essential part of a cluster, is crucial for characterizing system behavior at criticality.

Purpose of the Study:

  • To investigate the scaling behavior of the average backbone mass and its probability distribution in a 2D lattice at the percolation threshold.
  • To determine the critical exponents governing these scaling laws.

Main Methods:

  • Analysis of the backbone connecting two sites in a 2D lattice of size L at the percolation threshold.
  • Derivation of scaling forms for the average backbone mass and its probability distribution P(M(B)).

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Main Results:

  • The average backbone mass exhibits a scaling form ~ L^(dB)G(r/L), where G(x) ~ x^psi for x=r/L.
  • The exponent psi was found to be 0.37 ± 0.02, and the probability distribution P(M(B)) follows a power law M(B)^(-tauB) for small r, with tauB ≈ 1.20 ± 0.03.
  • The exponents satisfy the relation psi = dB(tauB-1), with psi representing the codimension of the backbone.

Conclusions:

  • The study provides a detailed characterization of backbone properties in 2D lattices at the percolation threshold.
  • The identified scaling laws and exponents offer insights into the fractal nature and connectivity of critical systems.