Asymptotic behavior of lifetime sums for random simplicial complex processes

TL;DR

This paper investigates the asymptotic behavior of lifetime sums in random simplicial complexes, extending classical graph results to higher dimensions and providing new bounds on Betti numbers related to Laplacian eigenvalues.

Contribution

It introduces a higher-dimensional analogue of Frieze's limit theorem, solves open questions on Linial-Meshulam and clique complexes, and develops a new Betti number bound based on Laplacian eigenvalues.

Findings

Asymptotic behavior of lifetime sums established

Solved open questions on specific complex processes

Derived new bounds on Betti numbers using Laplacian eigenvalues

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 $\zeta(3)$-limit theorem for the Erd\H{o}s-R\'{e}nyi 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.