二维Sierpinski垫片的生成树结构
Structure of spanning trees on the two-dimensional Sierpinski gasket
摘要：
考虑二维Sierpinski垫片SG（n）上的生成树，其中阶段$n$是非负整数。对于SG（n）的任意给定顶点$x$，我们严格地导出了顶点处度$j \in \{1,2,3,4\}$及其值在无限$n$极限内的概率分布.将所有顶点的概率除以顶点数，我们得到度的平均概率分布$j$。相应的极限分布$ϕ_j$给出了在所有生成树配置中顶点被1、2、3或4个键连接的平均概率。它们是有理数，表示为$ϕ_1=10957/40464$、$ϕ_2=6626035/13636368$、$ϕ_3=2943139/13636368$、$ϕ_4=124895/4545456$。
英文摘要：
Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage $n$ is a non-negative integer. For any given vertex $x$ of SG(n), we derive rigorously the probability distribution of the degree $j \in \{1,2,3,4\}$ at the vertex and its value in the infinite $n$ limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree $j$. The corresponding limiting distribution $ϕ_j$ gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as $ϕ_1=10957/40464$, $ϕ_2=6626035/13636368$, $ϕ_3=2943139/13636368$, $ϕ_4=124895/4545456$.
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