Journal of the Mathematical Society of Japan

Asymptotic behavior of lifetime sums for random simplicial complex processes

Masanori HINO and Shu KANAZAWA

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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 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.


This study was supported by JSPS KAKENHI Grant Number JP15H03625.

Article information

J. Math. Soc. Japan, Volume 71, Number 3 (2019), 765-804.

Received: 5 February 2018
First available in Project Euclid: 24 April 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 55U10: Simplicial sets and complexes 05E45: Combinatorial aspects of simplicial complexes 60C05: Combinatorial probability

Linial–Meshulam complex process random clique complex process multi-parameter random simplicial complex lifetime sum Betti number


HINO, Masanori; KANAZAWA, Shu. Asymptotic behavior of lifetime sums for random simplicial complex processes. J. Math. Soc. Japan 71 (2019), no. 3, 765--804. doi:10.2969/jmsj/79777977.

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