Xun Zhipeng, Hao Dapeng, Ziff Robert M
School of Material Sciences and Physics, China University of Mining and Technology, Xuzhou 221116, China.
Center for the Study of Complex Systems and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2800, USA.
Phys Rev E. 2021 Feb;103(2-1):022126. doi: 10.1103/PhysRevE.103.022126.
By means of extensive Monte Carlo simulation, we study extended-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors up to the eighth nearest neighbors for the square lattice and the ninth nearest neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods of connected sites can be mapped to problems of lattice percolation of extended objects of a given shape, such as disks and spheres, and the thresholds can be related to the continuum thresholds η_{c} for objects of those shapes. This mapping implies zp_{c}∼4η_{c}=4.51235 in two dimensions and zp_{c}∼8η_{c}=2.7351 in three dimensions for large z for circular and spherical neighborhoods, respectively, where z is the coordination number. Fitting our data for compact neighborhoods to the form p_{c}=c/(z+b) we find good agreement with this prediction, c=2^{d}η_{c}, with the constant b representing a finite-z correction term. We also examined results from other studies using this fitting formula. A good fit of the large but finite-z behavior can also be made using the formula p_{c}=1-exp(-2^{d}η_{c}/z), a generalization of a formula of Koza, Kondrat, and Suszcayński [J. Stat. Mech.: Theor. Exp. (2014) P110051742-546810.1088/1742-5468/2014/11/P11005]. We also study power-law fits which are applicable for the range of values of z considered here.
通过广泛的蒙特卡罗模拟，我们研究了正方形晶格和简单立方晶格上的扩展范围位点渗流，其中正方形晶格的最近邻组合包括直至第八近邻，简单立方晶格的最近邻组合包括直至第九近邻。我们使用单簇生长算法找到了23个系统的精确阈值。具有连通位点紧凑邻域的晶格上的位点渗流可以映射到给定形状的扩展对象（如圆盘和球体）的晶格渗流问题，并且这些阈值可以与这些形状对象的连续阈值ηₑ相关。对于圆形和球形邻域，这种映射分别意味着在二维中，对于大的z，zpₑ∼4ηₑ = ４.51235，在三维中，zpₑ∼8ηₑ = 2.7351，其中z是配位数。将我们紧凑邻域的数据拟合为pₑ = c/(z + b)的形式，我们发现与该预测有很好的一致性，c = 2ᵈηₑ，常数b代表有限z校正项。我们还使用这个拟合公式检查了其他研究的结果。使用公式pₑ = 1 - exp(-2ᵈηₑ/z)也可以很好地拟合大但有限z的行为，该公式是Koza、Kondrat和Suszcayński [《统计力学：理论与实验》（2014年）P110051742 - 546810.1088/1742 - 5468/2014/11/P11005]公式的推广。我们还研究了适用于此处考虑的z值范围的幂律拟合。
