跨越密集城市的最小生成树
Minimum spanning trees across dense cities
摘要：
考虑~ \(n\)节点独立地分布在~ \(N\)个城市中，这些城市包含在单位正方形~ \(S\)中，根据分布~ \(f.\)每个城市被建模为包含在~ \(S\)中的~ \(r_n \times r_n\)正方形，~ \(MSTC_n\)表示包含所有~ \(n\)节点的最小生成树的长度。我们使用近似的方法来获得方差估计的~ \(MSTC_n\)，并证明，如果城市是良好的连接和人口稠密在一定意义上，然后~ \(MSTC_n\)适当的中心和缩放收敛到零的概率。利用证明技巧，我们交替地导出了当节点独立地分布在单位正方形\(S.\)上时，最小生成树的长度\(MST_n\)的相应结果。我们得到了~ \(MST_n\)的方差至多是~ \(n\)的对数的幂，并利用一个子序列变元得到了~的几乎必然收敛\(MST_n\)适当居中和缩放。
英文摘要：
Consider~\(n\) nodes distributed independently across~\(N\) cities contained with the unit square~\(S\) according to a distribution~\(f.\) Each city is modelled as an~\(r_n \times r_n\) square contained within~\(S\) and~\(MSTC_n\) denotes the length of the minimum spanning tree containing all the~\(n\) nodes. We use approximation methods to obtain variance estimates for~\(MSTC_n\) and prove that if the cities are well-connected and densely populated in a certain sense, then~\(MSTC_n\) appropriately centred and scaled converges to zero in probability. Using the proof techniques, we alternately derive corresponding results for the length~\(MST_n\) of the minimum spanning tree for the usual case when the nodes are independently distributed throughout the unit square~\(S.\) In particular, we obtain that the variance of~\(MST_n\) grows at most as a power of the logarithm of~\(n\) and use a subsequence argument to get almost sure convergence of~\(MST_n\) appropriately centred and scaled.
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