Linial-Meshulam过程的最小生成代数和持久同调的寿命
Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process
摘要：
本文研究Frieze的$ζ(3)$ -极限定理在Erdös-Rényi图过程中的高维推广.弗里兹定理指出，当顶点数趋于无穷大时，最小生成树的期望权重收敛于$ζ(3)$。本文研究了随机单纯复形的模型$d$ -Linial-Meshulam过程，其中$d=1$对应于Erdös-Rényi图过程.首先，我们将生成无圈定义为生成树的高维类似物，并将其最小权与持久同调联系起来。然后，我们的主要结果表明，期望的最小生成acc的重量表现在$O(n^{d-1})$。
英文摘要：
This paper studies a higher dimensional generalization of Frieze's $ζ(3)$-limit theorem in the Erdös-Rényi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to $ζ(3)$ as the number of vertices goes to infinity. In this paper, we study the $d$-Linial-Meshulam process as a model for random simplicial complexes, where $d=1$ corresponds to the Erdös-Rényi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in $O(n^{d-1})$.
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