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–Renyi 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.