Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree
Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree
复制标题
凯莱树上键渗透关键阶段的局部簇大小统计
DOI：
10.1088/1742-5468/2016/05/053202
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Koji Nemoto
中科院分区：
文献类型：
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作者：
Tomoaki Nogawa;Takehisa Hasegawa;Koji Nemoto
我们研究键渗透的凯莱树（CT）通过专注于概率分布函数（PDF）的局部变量，即，包括选定的顶点的集群的大小。由于CT不具有不受边界效应影响的主导体区域，即使在大尺寸极限下，系统在其上的相位也不能很好地定义。本文中，我们表明，本地观察是有用的，以定义这样一个系统的相关联的系统上的Bethe格，即，一个无限的正则树没有边界的定义良好的相位。在逾渗阈值以上，CT中心的顶点（原点）和CT边界附近的顶点（叶子）的PDF具有不同的形式，这也不同于在欧几里得晶格的普通扩散相中观察到的PDF。CT起源的PDF是双峰的：衰减指数函数和系统大小相关的非对称峰，其服从具有分形指数的有限尺寸标度律。这些模式分别与有限和无限团簇在Bethe晶格的非唯一性相的PDF有关。另一方面，CT的叶的PDF是衰减的幂函数。这类似于在欧几里得晶格的临界点处观察到的PDF，但归因于CT在边界周围的嵌套结构。
We study bond percolation of the Cayley tree (CT) by focusing on the probability distribution function (PDF) of a local variable, namely, the size of the cluster including a selected vertex. Because the CT does not have a dominant bulk region, which is free from the boundary effect, even in the large-size limit, the phase of the system on it is not well defined. We herein show that local observation is useful to define the phase of such a system in association with the well-defined phase of the system on the Bethe lattice, that is, an infinite regular tree without boundary. Above the percolation threshold, the PDFs of the vertex at the center of the CT (the origin) and of the vertices near the boundary of the CT (the leaves) have different forms, which are also dissimilar to the PDF observed in the ordinary percolating phase of a Euclidean lattice. The PDF for the origin of the CT is bimodal: a decaying exponential function and a system-size-dependent asymmetric peak, which obeys a finite-size-scaling law with a fractal exponent. These modes are respectively related to the PDFs of the finite and infinite clusters in the nonuniqueness phase of the Bethe lattice. On the other hand, the PDF for the leaf of the CT is a decaying power function. This is similar to the PDF observed at a critical point of a Euclidean lattice but is attributed to the nesting structure of the CT around the boundary.