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Empirical equation for determining critical frontiers of mixed site-bond percolation in Archimedean lattices

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  • Lebrecht, W.

Abstract

An empirical polynomial equation ps(pb)=A+Kpbn is proposed to describe the behavior of critical frontiers for mixed site-bond percolation systems. This equation successfully recovers the logarithmic equation proposed by Yanuka & Englman, as well as the hyperbolic equation proposed by Tarasevich & van der Marck. For the latter, an analytical development is proposed using the logical operations provided by Tsallis on the S∩B and S∪B phases. The empirical equation proposed for n=4 successfully describes the critical S∪B frontier for square, hexagonal, triangular and Kagome lattices. An extension of this equation is performed for simple systems linked as dimers in square and triangular lattices.

Suggested Citation

  • Lebrecht, W., 2025. "Empirical equation for determining critical frontiers of mixed site-bond percolation in Archimedean lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 661(C).
    • Handle: RePEc:eee:phsmap:v:661:y:2025:i:c:s0378437125000524
    DOI: 10.1016/j.physa.2025.130400
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    1. González, M.I. & Centres, P. & Lebrecht, W. & Ramirez-Pastor, A.J. & Nieto, F., 2013. "Site–bond percolation on triangular lattices: Monte Carlo simulation and analytical approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(24), pages 6330-6340.
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    5. M. Dolz & F. Nieto & A. J. Ramirez-Pastor, 2005. "Dimer site-bond percolation on a square lattice," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 43(3), pages 363-368, February.
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    7. Tsallis, Constantino, 2004. "Some thoughts on theoretical physics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(3), pages 718-736.
    8. Lebrecht, W. & Centres, P.M. & Ramirez-Pastor, A.J., 2019. "Analytical approximation of the site percolation thresholds for monomers and dimers on two-dimensional lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 516(C), pages 133-143.
    9. Lebrecht, W. & Centres, P.M. & Ramirez-Pastor, A.J., 2021. "Empirical formula for site and bond percolation thresholds on Archimedean and 2-uniform lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 569(C).
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