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 neighborhoods. The dependence of the value of the percolation thresholds $p_c$ on the coordination number $z$ are tested against various theoretical predictions. The proposed single scalar index $\xi ={\sum }_i{z}_i{r}_i^2/i$ (depending on the coordination zone number $i$, the neighborhood coordination number $z$, and the square distance $r^2$ to sites in $i\mathrm{th}$ coordination zone from the central site) allows one to differentiate among various neighborhoods and relate $p_c$ to $\xi$. The thresholds roughly follow a power law $p_c\propto {\xi }^{-\gamma }$ with $\gamma \approx 0.710\left(19\right)$.

中文翻译：

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三角形网格上的渗漏阈值，用于包含直至第五个协调区的站点的邻域

我们确定阈值 $p_C$对于包含从第一个到第五个协调区域的站点（包括它们的复杂组合）的所有可用邻域，在三角晶格上进行随机站点渗透。有31个不同的社区。渗透阈值的依赖性$p_C$ 在协调号上 $ž$针对各种理论预测进行了测试。拟议的单标量指数$\xi ={\sum }_{\mathrm{一世}}{ž}_{\mathrm{一世}}{\mathrm{[R}}_{\mathrm{一世}}^{\mathrm{2个}}/\mathrm{一世}$ （取决于协调区编号 $\mathrm{一世}$，邻里协调号码 $ž$，以及平方距离 ${\mathrm{[R}}^{\mathrm{2个}}$ 前往的网站 $\mathrm{一世}日$ 中心区域的协调区）允许人们在各个邻里之间进行区分并建立联系 $p_C$ 至 $\xi$。阈值大致遵循幂定律$p_C\propto {\xi }^{-\gamma }$ 和 $\gamma \approx 0.710（19）$。