Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 71, Number 3 (2019), 765-804.
Asymptotic behavior of lifetime sums for random simplicial complex processes
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.
J. Math. Soc. Japan, Volume 71, Number 3 (2019), 765-804.
Received: 5 February 2018
First available in Project Euclid: 24 April 2019
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.jmsj/1556092819