Site percolation thresholds for Archimedean lattices

Paul N. Suding; Robert M. Ziff

doi:10.1103/PhysRevE.60.275

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Site percolation thresholds for Archimedean lattices

Paul N. Suding and Robert M. Ziff^*

Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136

^*Electronic address: rziff@engin.umich.edu

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Phys. Rev. E 60, 275 – Published 1 July, 1999

DOI: https://doi.org/10.1103/PhysRevE.60.275

Abstract

Precise thresholds for site percolation on eight Archimedean lattices are determined by the hull-walk gradient-percolation simulation method, with the results $p_{c} = 0.697 043,$ honeycomb or ${(6}^{3}),$ 0.807 904 ${(3, 12}^{2}),$ 0.747 806 (4,6,12), 0.729 724 ${(4, 8}^{2}),$ 0.579 498 ${(3}^{4}, 6),$ 0.621 819 (3,4,6,4), 0.550 213 ${(3}^{3} {, 4}^{2}),$ and 0.550 806 ${(3}^{2}, 4, 3, 4),$ with errors of about $\pm 3 \times 10^{- 6} .$ [The remaining Archimedean lattices are the square ${(4}^{4}),$ triangular ${(3}^{6}),$ and Kagomé (3,6,3,6), for which $p_{c}$ is already known exactly or to a high degree of accuracy.] The numerical result for the ${(3, 12}^{2})$ lattice is consistent with the exact value $[1 - 2 \sin (π / 18)]^{1 / 2} .$ The values of $p_{c}$ for all 11 Archimedean lattices, as well as a number of nonuniform lattices, are found to be well correlated by a nearly linear function of a generalized Scher-Zallen filling factor. This correlation is much more accurate than recently proposed correlations based solely upon coordination number.

References (47)

J. V. Field, Vistas Astron. 23, 109 (1979); Arch. Hist. Exact Sci. 50, 241 (1997).
L. Fejes Tóth, Acta Math. 11, 363 (1960).
F. C. Frank and J. S. Kasper, Acta Crystallogr. 11, 184 (1958); ibid.12, 483 (1959).
H. Takeda and J. D. H. Donnay, Acta Crystallogr. 19, 474 (1965).
R. Shrock and S. H. Tsai, Phys. Rev. E 56, 2733, 4111 (1997).
C. d’Iribarne, G. Rasigni, and M. Rasigni, Phys. Lett. A 209, 95 (1995).
C. Dussert, G. Rasigni, and M. Rasigni, Phys. Lett. A 139, 35 (1989).
S. C. van der Marck, Phys. Rev. E 55, 1514 (1997); ibid.56, 3732(E) (1997).
S. C. van der Marck, Int. J. Mod. Phys. C 9, 529 (1998).
T. Vicsek and J. Kertész, Phys. Lett. 81A, 51 (1981).
Z. V. Djordjevic, H. E. Stanley, and A. Margolina, J. Phys. A 15, L405 (1982).
B. Sapoval, M. Rosso, and J. F. Gouyet, J. Phys. (France) Lett. 46, L149 (1985).
M. Rosso, J. F. Gouyet, and B. Sapoval, Phys. Rev. B 32, 6053 (1986).
R. M. Ziff, P. T. Cummings, and G. Stell, J. Phys. A 17, 3009 (1984).
J. W. Essam and M. F. Sykes, J. Math. Phys. 5, 1117 (1964).
R. M. Ziff, Phys. Rev. Lett. 69, 2670 (1992).
S. Galam and A. Mauger, J. Appl. Phys. 75, 5526 (1994).
S. Galam and A. Mauger, Physica A 205, 502 (1994).
S. Galam and A. Mauger, Phys. Rev. E 53, 2177 (1996).
S. Galam and A. Mauger, Phys. Rev. E 55, 1230 (1997).
S. Galam and A. Mauger, Phys. Rev. E 56, 322 (1997).
S. C. van der Marck, Phys. Rev. E 55, 1228 (1997).
S. C. van der Marck, Phys. Rev. E 55, 6593 (1997).
S. C. van der Marck, J. Phys. A 31, 3449 (1998).
B. Grünbaum and G. C. Shephard, Tilings and Patterns (Freeman, New York, 1987).
R. M. Ziff and B. Sapoval, J. Phys. A 18, L1169 (1986).
R. M. Ziff and G. Stell (unpublished).
R. M. Ziff and P. N. Suding, J. Phys. A 30, 5351 (1997).
D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor and Francis, London, 1994).
A. Weinrib and S. Trugman, Phys. Rev. B 31, 2993 (1985).
R. M. Ziff, Comput. Phys. 12, 385 (1998).
S. W. Golomb, Shift Register Sequences (Holden-Day, San Francisco, 1967).
N. Zierler, Inform. Control 15, 67 (1969).
P. Préa, Math. Comput. Modeling 26, 317 (1997).
I. Jensen and A. J. Guttmann, J. Phys. A 31, 8137 (1998).
S. Galam and A. Mauger, Eur. Phys. J. B1, 255 (1998).
F. Babalievski, Phys. Rev. E 59, 1278 (1999).
H. Scher and R. Zallen, J. Chem. Phys. 53, 3759 (1970).
F. Yonezawa, S. Sakamoto, and M. Hori, Phys. Rev. B 40, 636 (1989).
F. Yonezawa, S. Sakamoto, and M. Hori, Phys. Rev. B 40, 650 (1989).
S. C. van der Marck (private communication).
J. C. Wierman, J. Phys. A 17, 1525 (1984); ibid.21, 1487 (1988).
J. P. Lu and J. L. Birman, J. Stat. Phys. 46, 1057 (1987).
R. M. Ziff and F. Babalievski, Physica A (to be published).
G. Ord and S. G. Whittington, J. Phys. A 13, L307 (1980); ibid.15, L29 (1982).
R. M. Ziff, Int. J. Mod. Phys. C (to be published).
C. d’Iribarne, M. Rasigni, and G. Rasigni, J. Phys. A 32, 2611 (1999).

[c1] J. V. Field, Vistas Astron. 23, 109 (1979); Arch. Hist. Exact Sci. 50, 241 (1997).

[c2] L. Fejes Tóth, Acta Math. 11, 363 (1960).

[c3] F. C. Frank and J. S. Kasper, Acta Crystallogr. 11, 184 (1958); ibid.12, 483 (1959).

[c4] H. Takeda and J. D. H. Donnay, Acta Crystallogr. 19, 474 (1965).

[c5] R. Shrock and S. H. Tsai, Phys. Rev. E 56, 2733, 4111 (1997).

[c6] C. d’Iribarne, G. Rasigni, and M. Rasigni, Phys. Lett. A 209, 95 (1995).

[c7] C. Dussert, G. Rasigni, and M. Rasigni, Phys. Lett. A 139, 35 (1989).

[c8] S. C. van der Marck, Phys. Rev. E 55, 1514 (1997); ibid.56, 3732(E) (1997).

[c9] S. C. van der Marck, Int. J. Mod. Phys. C 9, 529 (1998).

[c10] T. Vicsek and J. Kertész, Phys. Lett. 81A, 51 (1981).

[c11] Z. V. Djordjevic, H. E. Stanley, and A. Margolina, J. Phys. A 15, L405 (1982).

[c12] B. Sapoval, M. Rosso, and J. F. Gouyet, J. Phys. (France) Lett. 46, L149 (1985).

[c13] M. Rosso, J. F. Gouyet, and B. Sapoval, Phys. Rev. B 32, 6053 (1986).

[c14] R. M. Ziff, P. T. Cummings, and G. Stell, J. Phys. A 17, 3009 (1984).

[c15] J. W. Essam and M. F. Sykes, J. Math. Phys. 5, 1117 (1964).

[c16] R. M. Ziff, Phys. Rev. Lett. 69, 2670 (1992).

[c17] S. Galam and A. Mauger, J. Appl. Phys. 75, 5526 (1994).

[c18] S. Galam and A. Mauger, Physica A 205, 502 (1994).

[c19] S. Galam and A. Mauger, Phys. Rev. E 53, 2177 (1996).

[c20] S. Galam and A. Mauger, Phys. Rev. E 55, 1230 (1997).

[c21] S. Galam and A. Mauger, Phys. Rev. E 56, 322 (1997).

[c22] S. C. van der Marck, Phys. Rev. E 55, 1228 (1997).

[c23] S. C. van der Marck, Phys. Rev. E 55, 6593 (1997).

[c24] S. C. van der Marck, J. Phys. A 31, 3449 (1998).

[c25] B. Grünbaum and G. C. Shephard, Tilings and Patterns (Freeman, New York, 1987).

[c26] R. M. Ziff and B. Sapoval, J. Phys. A 18, L1169 (1986).

[c27] R. M. Ziff and G. Stell (unpublished).

[c28] R. M. Ziff and P. N. Suding, J. Phys. A 30, 5351 (1997).

[c29] D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor and Francis, London, 1994).

[c30] A. Weinrib and S. Trugman, Phys. Rev. B 31, 2993 (1985).

[c31] R. M. Ziff, Comput. Phys. 12, 385 (1998).

[c32] S. W. Golomb, Shift Register Sequences (Holden-Day, San Francisco, 1967).

[c33] N. Zierler, Inform. Control 15, 67 (1969).

[c34] P. Préa, Math. Comput. Modeling 26, 317 (1997).

[c35] I. Jensen and A. J. Guttmann, J. Phys. A 31, 8137 (1998).

[c36] S. Galam and A. Mauger, Eur. Phys. J. B1, 255 (1998).

[c37] F. Babalievski, Phys. Rev. E 59, 1278 (1999).

[c38] H. Scher and R. Zallen, J. Chem. Phys. 53, 3759 (1970).

[c39] F. Yonezawa, S. Sakamoto, and M. Hori, Phys. Rev. B 40, 636 (1989).

[c40] F. Yonezawa, S. Sakamoto, and M. Hori, Phys. Rev. B 40, 650 (1989).

[c41] S. C. van der Marck (private communication).

[c42] J. C. Wierman, J. Phys. A 17, 1525 (1984); ibid.21, 1487 (1988).

[c43] J. P. Lu and J. L. Birman, J. Stat. Phys. 46, 1057 (1987).

[c44] R. M. Ziff and F. Babalievski, Physica A (to be published).

[c45] G. Ord and S. G. Whittington, J. Phys. A 13, L307 (1980); ibid.15, L29 (1982).

[c46] R. M. Ziff, Int. J. Mod. Phys. C (to be published).

[c47] C. d’Iribarne, M. Rasigni, and G. Rasigni, J. Phys. A 32, 2611 (1999).

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Site percolation thresholds for Archimedean lattices

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