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A percolation system with extremely long range connections and node dilution

Author

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  • de Almeida, M.L.
  • Albuquerque, E.L.
  • Fulco, U.L.
  • Serva, M.

Abstract

We study the very long-range bond-percolation problem on a linear chain with both node and bond dilution. Very long-range means that the probability pij for a connection between two nodes i,j at a distance rij decays as a power-law, i.e. pij=ρ/[rijαN1−α] when α∈[0,1), and pij=ρ/[rijln(N)] when α=1. Node dilution means that the probability that a node is present in a site is ps∈(0,1]. The behavior of this model results from the competition between long-range connectivity which enhances the percolation, and node dilution which weakens percolation. The case α=0 with ps=1 is well-known, being the exactly solvable mean-field model. The percolation order parameter P∞ is investigated numerically for different values of α, ps and ρ. We show that in all range α∈[0,1] the percolation order parameter P∞ depends only on the average connectivity γ of the nodes, which can be explicitly computed in terms of the three parameters α, ps and ρ.

Suggested Citation

  • de Almeida, M.L. & Albuquerque, E.L. & Fulco, U.L. & Serva, M., 2014. "A percolation system with extremely long range connections and node dilution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 416(C), pages 273-278.
    • Handle: RePEc:eee:phsmap:v:416:y:2014:i:c:p:273-278
    DOI: 10.1016/j.physa.2014.08.041
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    References listed on IDEAS

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    1. Zhang, Zhongyang & Zhang, Lixin, 2013. "Scaling limits for one-dimensional long-range percolation: Using the corrector method," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2459-2466.
    2. Serva, Maurizio, 2011. "Exact and approximate solutions for the dilute Ising model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(13), pages 2443-2451.
    3. Serva, Maurizio, 2010. "Magnetization densities as replica parameters: The dilute ferromagnet," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2700-2707.
    4. Rego, Henio H.A & Lucena, Liacir S & da Silva, Luciano R & Tsallis, Constantino, 1999. "Crossover from extensive to nonextensive behavior driven by long-range d=1 bond percolation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 266(1), pages 42-48.
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