论文标题
键盘渗透的簇大小在柏拉图固体上
Cluster size in bond percolation on the Platonic solids
论文作者
论文摘要
本文的主要目的是研究五个柏拉图固体的典型键渗透簇的大小：四面体，立方体，八面体，十二面体和二十面体。从动态的角度来看群集，即将簇与出生过程进行比较，我们首先证明了群集大小的第一矩和第二矩在某个分支过程中由它们的对应物界定，这会导致明确的上限，当开放边缘的密度很小时，这些上限是准确的。使用封闭边缘所包围的顶点无法通过开放路径来达到，我们还得出了上限，相反，当开放边缘的密度较大时，这是准确的。实际上，这些上限对于所有常规图。专门研究五个柏拉图固体，第一和第二矩的确切值是从包含 - 排斥原理和计算机程序中获得的。我们程序的目的不是模拟随机过程，而是要精确地计算出太大而无法手动计算的整数总和，以便这些结果是分析性的，而不是数值的。
The main objective of this paper is to study the size of a typical cluster of bond percolation on each of the five Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. Looking at the clusters from a dynamical point of view, i.e., comparing the clusters with birth processes, we first prove that the first and second moments of the cluster size are bounded by their counterparts in a certain branching process, which results in explicit upper bounds that are accurate when the density of open edges is small. Using that vertices surrounded by closed edges cannot be reached by an open path, we also derive upper bounds that, on the contrary, are accurate when the density of open edges is large. These upper bounds hold in fact for all regular graphs. Specializing in the five~Platonic solids, the exact value of (or lower bounds for) the first and second moments are obtained from the inclusion-exclusion principle and a computer program. The goal of our program is not to simulate the stochastic process but to compute exactly sums of integers that are too large to be computed by hand so these results are analytical, not numerical.
