Universal condition for critical percolation thresholds of kagomé-like lattices.

@article{Ziff2008UniversalCF,
  title={Universal condition for critical percolation thresholds of kagom{\'e}-like lattices.},
  author={Robert M. Ziff and Hang Gu},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2008},
  volume={79 2 Pt 1},
  pages={
          020102
        }
}
  • R. ZiffH. Gu
  • Published 1 December 2008
  • Computer Science
  • Physical review. E, Statistical, nonlinear, and soft matter physics
Lattices that can be represented in a kagomé-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between P3 , the probability that all three vertices in the triangle connect, and P0 , the probability that none connect. A linear approximation for P3(P0) is derived and appears to provide a rigorous upper bound for critical thresholds. A numerically determined relation for P3(P0) gives thresholds for the kagomé, site-bond honeycomb, (3-12;{2… 
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References

SHOWING 1-10 OF 38 REFERENCES

Introduction To Percolation Theory

Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in…

Combin

  • Probab. Comput. 14, 549
  • 2005

Eur. Phys. J. B

  • Eur. Phys. J. B
  • 2007

Phys. Rev. E

  • Phys. Rev. E
  • 2008

J. Phys. A

  • J. Phys. A
  • 1986

Phys

  • Rev. E 73, 016134
  • 2006

J. Stat. Phys

  • J. Stat. Phys
  • 2006

Combin. Probab. Comput

  • Combin. Probab. Comput
  • 2005

Phys

  • Rev. Lett. 96, 090602
  • 2006

Phys

  • Rev. B 32, 6053
  • 1985