Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process

Abstract

This paper studies a higher dimensional generalization of Frieze's $\zeta (3)$-limit theorem in the Erdös-Rényi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to $\zeta (3)$ 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 $O(n^{d-1})$.

24 pages

Cited by in corpus (1)

• Limit theorems for persistence diagrams