Monte Carlo simulation of bond percolation on square lattice with complex neighborhoods

Abstract

Based on an effective single cluster growth algorithm, bond percolation on square lattice with the nearest neighbors, the next nearest neighbors, up to the 5th nearest neighbors are investigated by Monte Carlo simulations. The bond percolation thresholds for more than 20 lattices are deduced, and the correlations between percolation threshold <inline-formula><tex-math id="M8">\begin{document}$p_{\rm c}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M8.png"/></alternatives></inline-formula> and lattice structures are discussed in depth. By introducing the index <inline-formula><tex-math id="M9">\begin{document}$\xi = \displaystyle\sum\nolimits_{i} z_{i} r_{i}^{2} / i$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M9.png"/></alternatives></inline-formula> to remove the degeneracy, it is found that the thresholds follow a power law <inline-formula><tex-math id="M10">\begin{document}$p_{\rm c} \propto \xi^{-\gamma}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M10.png"/></alternatives></inline-formula>, with <inline-formula><tex-math id="M11">\begin{document}$\gamma \approx 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M11.png"/></alternatives></inline-formula>, where <inline-formula><tex-math id="M12">\begin{document}$z_{i}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M12.png"/></alternatives></inline-formula> is the <i>i</i>th neighborhood coordination number, and <inline-formula><tex-math id="M13">\begin{document}$r_{i}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20211757_M13.png"/></alternatives></inline-formula> is the distance between sites in the <i>i</i>-th coordination zone and the central site.