影子知道：随机复形的最小生成无圈与持续图的经验分布
The Shadow knows: Empirical Distributions of Minimum Spanning Acycles and Persistence Diagrams of Random Complexes
AI总结：
本文将 Frieze 关于随机 MST 权值和的结果推广，利用“影子”刻画随机 MST 与 MSA 全部权重的经验分布极限，并推出加权单纯复形持续图死亡时间的相应结论。
摘要：
1985年，Frieze证明了均匀加权图中最小生成树（MST）边权的期望和收敛于 ζ(3)。最近，Hino 和 Kanazawa 将这一结果推广到均匀加权单纯复形，其中 MST 的角色由其高维类似物——最小生成无圈（MSA）——承担。我们的工作更进一步，描述了该随机 MST 和随机 MSA 中所有权重的直方图。具体而言，我们证明它们的经验分布收敛到一种基于“影子”概念的测度。图的影子是所有缺失传递边的集合；而对于单纯复形，它是一种相关的拓扑推广。作为推论，我们对与上述加权复形对应的持续图中的死亡时间得到了类似结论，这是应用拓扑学中一项受关注的结果。
英文摘要：
In 1985, Frieze showed that the expected sum of the edge weights of the minimum spanning tree (MST) in the uniformly weighted graph converges to $ζ(3)$. Recently, Hino and Kanazawa extended this result to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analog -- the Minimum Spanning Acycle (MSA). Our work goes beyond and describes the histogram of all the weights in this random MST and random MSA. Specifically, we show that their empirical distributions converge to a measure based on a concept called the shadow. The shadow of a graph is the set of all the missing transitive edges, and, for a simplicial complex, it is a related topological generalization. As a corollary, we obtain a similar claim for the death times in the persistence diagram corresponding to the above-weighted complex, a result of interest in applied topology.
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