Malarz Krzysztof
AGH University, Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Kraków, Poland.
Phys Rev E. 2024 Mar;109(3-1):034108. doi: 10.1103/PhysRevE.109.034108.
The phenomenon of percolation is one of the core topics in statistical mechanics. It allows one to study the phase transition known in real physical systems only in a purely geometrical way. In this paper, we determine thresholds p_{c} for random-site percolation in triangular and honeycomb lattices for all available neighborhoods containing sites from the sixth coordination zone. The results obtained (together with the percolation thresholds gathered from the literature also for other complex neighborhoods and also for a square lattice) show the power-law dependence p_{c}∝(ζ/K)^{-γ} with γ=0.526(11), 0.5439(63), and 0.5932(47), for a honeycomb, square, and triangular lattice, respectively, and p_{c}∝ζ^{-γ} with γ=0.5546(67) independently on the underlying lattice. The index ζ=∑{i}z{i}r_{i} stands for an average coordination number weighted by distance, that is, depending on the coordination zone number i, the neighborhood coordination number z_{i}, and the distance r_{i} to sites in the ith coordination zone from the central site. The number K indicates lattice connectivity, that is, K=3, 4, and 6 for the honeycomb, square, and triangular lattice, respectively.
渗流现象是统计力学的核心主题之一。它使人们能够仅以纯粹几何的方式研究实际物理系统中已知的相变。在本文中，对于包含来自第六配位区位点的所有可用邻域，我们确定了三角形和蜂窝晶格中随机位点渗流的阈值(p_{c})。所获得的结果（连同从文献中收集的其他复杂邻域以及正方形晶格的渗流阈值）表明，对于蜂窝晶格、正方形晶格和三角形晶格，幂律依赖关系分别为(p_{c}\propto(ζ/K)^{-γ})，其中(γ = 0.526(11))、(0.5439(63))和(0.5932(47))，并且与底层晶格无关，(p_{c}\proptoζ^{-γ})，其中(γ = 0.5546(67))。指数(ζ = \sum_{i}z_{i}r_{i})表示由距离加权的平均配位数，即取决于配位区编号(i)、邻域配位数(z_{i})以及从中心位点到第(i)个配位区中位点的距离(r_{i})。数字(K)表示晶格连通性，即对于蜂窝晶格、正方形晶格和三角形晶格，(K)分别为(3)、(4)和(6)。
