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On AB Bond Percolation on the Square Lattice and AB Site Percolation on Its Line Graph

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Abstract

We prove that AB site percolation occurs on the line graph of the square lattice when \(p \in (1 - \sqrt {1 - p_c } ,\sqrt {1 - p_c } )\), where p c is the critical probability for site percolation in \(\mathbb{Z}^2\). Also, we prove that AB bond percolation does not occur on \(\mathbb{Z}^2\) for p = \(\frac{1}{2}\).

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REFERENCES

  1. J. W. Halley, Percolation structures and processes, G. Deutscher, R. Zallen, and J. Adler, eds. (Bristol, Adam Hilger, 1983), pp. 323-351.

    Google Scholar 

  2. F. Sevsek, J. M. Debierre, and L. Turban, Antipercolation on Bethe and triangular lattices, J. Phys. A: Math. Gen. 16:801-810 (1983).

    Google Scholar 

  3. T. Mai and J. W. Halley, Ordering in two dimensions, S. Sinha, ed. (Amsterdam, North-Holland, 1980), pp. 369-371.

    Google Scholar 

  4. M. J. Appel and J. C. Wierman, On the absence of infinite AB percolation clusters in bipartite graphs, J. Phys. A: Math. Gen. 20:2527-2531 (1987).

    Google Scholar 

  5. E. R. Scheinerman and J. C. Wierman, Infinite AB clusters exist, J. Phys. A: Math. Gen. 20:1305-1307 (1987).

    Google Scholar 

  6. J. C. Wierman and M. J. Appel, Infinite AB percolation clusters exist on the triangular lattice, J. Phys. A: Math. Gen. 20:2533-2537 (1987).

    Google Scholar 

  7. G. R. Grimmett, Percolation, 2nd Ed. (Springer-Verlag, New York, 1999).

    Google Scholar 

  8. R. M. Ziff and B. Sapoval, The efficient determination of the percolation threshold by a frontier-generating walk in a gradient, J. Phys. A: Math. Gen 19:1169-1172 (1986).

    Google Scholar 

  9. J. C. Wierman, Substitution method critical probability bounds for the square lattice site percolation model, Combin. Probab. Comput. 4:181-188 (1995).

    Google Scholar 

  10. M. J. Appel, AB percolation on plane triangulations is unimodal, J. Appl. Probab. 31:193-204 (1994).

    Google Scholar 

  11. T. M. Liggett, R. H. Schonmann, and A. M. Stacey, Domination by product measures, Ann. Probab. 25:71-95 (1997).

    Google Scholar 

  12. G. R. Grimmett, Infinite paths in randomly oriented lattices, Random Structures Algorithms 18:257-266 (2001).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Capital Normal University, Beijing, People's Republic of China

    Xian-Yuan Wu

  2. Departamento de Estatística, Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão 1010, CEP 05508–090, São Paulo SP, Brasil

    S. Yu. Popov

Authors
  1. Xian-Yuan Wu
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  2. S. Yu. Popov
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Wu, XY., Popov, S.Y. On AB Bond Percolation on the Square Lattice and AB Site Percolation on Its Line Graph. Journal of Statistical Physics 110, 443–449 (2003). https://doi.org/10.1023/A:1021091316925

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  • DOI: https://doi.org/10.1023/A:1021091316925

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