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Title: Asymptotic behavior of lifetime sums for random simplicial complex processes
Authors: HINO, Masanori
KANAZAWA, Shu
Author's alias: 日野, 正訓
金澤, 秀
Keywords: Linial–Meshulam complex process
random clique complex process
multi-parameter random simplicial complex
lifetime sum
Betti number
Issue Date: Jul-2019
Publisher: Mathematical Society of Japan
Journal title: Journal of the Mathematical Society of Japan
Volume: 71
Issue: 3
Start page: 765
End page: 804
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 ζ(3)-limit theorem for the Erdős–Rényi 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.
Rights: © 2019 Mathematical Society of Japan
The full-text file will be made open to the public on 1 July 2022 in accordance with publisher's 'Terms and Conditions for Self-Archiving'.
URI: http://hdl.handle.net/2433/243268
DOI(Published Version): 10.2969/jmsj/79777977
Related Link: https://projecteuclid.org/euclid.jmsj/1556092819
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