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تسجيل مستخدم جديدPercolation thresholds on triangular lattice for neighbourhoods containing sites up-to the fifth coordination zone
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K. Malarz
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We determine thresholds $p_c$ for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighbourhoods. The dependence of the value of the percolation thresholds $p_c$ on the coordination number $z$ are tested against various theoretical predictions. The newly proposed single scalar index $xi=sum_i z_ir_i^2/i$ (depending on the coordination zone number $i$, the neighbourhood coordination number $z$ and the square-distance $r^2$ to sites in $i$-th coordination zone from the central site) allows to differentiate among various neighbourhoods and relate $p_c$ to $xi$. The thresholds roughly follow a power law $p_cproptoxi^{-gamma}$ with $gammaapprox 0.710(19)$.
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We determine thresholds $p_c$ for random site percolation on a triangular lattice for neighbourhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN) and next-next-next-next-nearest (5NN) neighbours,
and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [M. E. J. Newman and R. M. Ziff, Physical Review E 64, 016706 (2001)], for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with a method, by Bastas et al. [N. Bastas, K. Kosmidis, P. Giazitzidis, and M. Maragakis, Physical Review E 90, 062101 (2014)], of estimating thresholds from low statistics data. The estimated values of percolation thresholds are $p_c(text{4NN})=0.192410(43)$, $p_c(text{3NN+2NN})=0.232008(38)$, $p_c(text{5NN+4NN})=0.140286(5)$, $p_c(text{3NN+2NN+NN})=0.215484(19)$, $p_c(text{5NN+4NN+NN})=0.131792(58)$, $p_c(text{5NN+4NN+3NN+2NN})=0.117579(41)$, $p_c(text{5NN+4NN+3NN+2NN+NN})=0.115847(21)$. The method is tested on the standard case of site percolation on triangular lattice, where $p_c(text{NN})=p_c(text{2NN})=p_c(text{3NN})=p_c(text{5NN})=frac{1}{2}$ is recovered with five digits accuracy $p_c(text{NN})=0.500029(46)$ by averaging over one thousand lattice realisations only.
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