逾渗集团的几何结构.docVIP

逾渗集团的几何结构 徐枭1，王俊峰1，周宗政2，蒂姆 M 加罗尼2，邓友金1 1 中国科学技术大学合肥微尺度物质科学国家实验室及近代物理系，合肥　 230026 2 莫纳什大学数学科学学院，克莱顿，维多利亚 3800 澳大利亚 摘要：通过模拟周期性方形格上的键逾渗，我们研究了逾渗集团的几何性质。我们的研究显 示，对于大的系统尺寸，桥键和非桥键的密度均趋于 1/4。使用蒙特卡罗模拟，我们研究了格 子上一条给定的边不是一根桥键但它的两个环弧 (loop arcs) 属于同一个环的概率，并发现此 概率的行为被双臂指数 (two-arm exponent) 所描述。接着我们将桥键分为两类：树枝键和枢 纽键。一根桥键如果满足下述特征将被归类为树枝键：删除此键后所生成的两个集团中至少有 一个集团为树形集团。从普通的逾渗构型出发，删除掉所有的树枝键将形成一个 leaf-free 构 型，而删除掉所有的桥键将产生一个 bridge-free 构型。尽管树枝键数目在所有占据键数目中 的比例 ≈ 43%，leaf-free 集团及其外壳 (hull) 长度的分形维均与相应的标准的逾渗集团的分形 维相一致。相反，我们发现 bridge-free 构型上的集团尺寸以及 bridge-free 构型的外壳的尺寸 的分形维数分别由骨干维和外部周长维给出。我们估计骨干团簇分形维是 1.643 36(10)。 关键词：逾渗集团，几何结构，相变，临界现象，分形维 中图分类号： O414.2 Geometric structure of percolation clusters XU Xiao1, WANG Jun-Feng1, Zhou Zong-Zheng2, Timothy M. Garoni2, Deng You-Jin1 1 Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026 2 School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia Abstract: 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 ?nd that it is governed by the two-arm exponent. We then classify bridges into two types: branches and junctions. A bridge is a branch i? at least one of the two clusters produced by its deletion is a tree. Starting from a percolation con?guration and deleting the branches results in a leaf-free con?guration, while deleting all bridges produces a bridge-free con?guration. Although branches account for ≈ 43% of all occupied bonds, we ?nd that the fractal dimensions of the cluster size and hull length of leaf-free con?gurations are consistent 基 金 项 目： The Specialized Research Fund for the Doctoral Program of High