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.
"Asymptotic behavior of lifetime sums for random simplicial complex processes." J. Math. Soc. Japan 71 (3) 765 - 804, July, 2019. https://doi.org/10.2969/jmsj/79777977