Xu Xiao, Wang Junfeng, Zhou Zongzheng, Garoni Timothy M, Deng Youjin
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.
School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Jan;89(1):012120. doi: 10.1103/PhysRevE.89.012120. Epub 2014 Jan 15.
We investigate the geometric properties of percolation clusters by studying square-lattice bond percolation on the torus. We show that the density of bridges and nonbridges both tend to 1/4 for large system sizes. Using Monte Carlo simulations, we study the probability that a given edge is not a bridge but has both its loop arcs in the same loop and find that it is governed by the two-arm exponent. We then classify bridges into two types: branches and junctions. A bridge is a branch iff at least one of the two clusters produced by its deletion is a tree. Starting from a percolation configuration and deleting the branches results in a leaf-free configuration, whereas, deleting all bridges produces a bridge-free configuration. Although branches account for ≈43% of all occupied bonds, we find that the fractal dimensions of the cluster size and hull length of leaf-free configurations are consistent with those for standard percolation configurations. By contrast, we find that the fractal dimensions of the cluster size and hull length of bridge-free configurations are given by the backbone and external perimeter dimensions, respectively. We estimate the backbone fractal dimension to be 1.643 36(10).
我们通过研究环面上的方格晶格键渗流来探究渗流团簇的几何性质。我们表明，对于大系统规模，桥和非桥的密度都趋于1/4。利用蒙特卡罗模拟，我们研究了给定边不是桥但其两个环弧在同一个环中的概率，并发现它由双臂指数控制。然后我们将桥分为两类：分支和节点。如果删除一条桥所产生的两个团簇中至少有一个是树，那么这条桥就是一个分支。从一个渗流构型开始并删除分支会得到一个无叶构型，而删除所有桥会得到一个无桥构型。尽管分支占所有被占据键的约43%，但我们发现无叶构型的团簇大小和包壳长度的分形维数与标准渗流构型的一致。相比之下，我们发现无桥构型的团簇大小和包壳长度的分形维数分别由主链维和外部周长维数给出。我们估计主链分形维数为1.643 36(10)。
