Asymptotic behavior of lifetime sums for random simplicial complex processes

Masanori Hino; Shu Kanazawa

doi:10.2969/jmsj/79777977

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arXiv · 1802.00548

Asymptotic behavior of lifetime sums for random simplicial complex processes

Masanori Hino·Shu Kanazawa

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Abstract

We study the homological properties of random simplicial complexes. In particular, we obtain the asymptotic behavior of lifetime sums for a class of increasing random simplicial complexes; this result is a higher-dimensional counterpart of Frieze's $\zeta(3)$-limit theorem for the Erd\H{o}s-R\'{e}nyi graph process. The main results include solutions to questions posed in an earlier study by Hiraoka and Shirai about the Linial-Meshulam complex process and the random clique complex process. One of the key elements of the arguments is a new upper bound on the Betti numbers of general simplicial complexes in terms of the number of small eigenvalues of Laplacians on links. This bound can be regarded as a quantitative version of the cohomology vanishing theorem.

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Masanori Hino, Shu Kanazawa. 2018-02-02. Asymptotic behavior of lifetime sums for random simplicial complex processes. https://doi.org/10.2969/jmsj%2F79777977

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