Law of large numbers for Betti numbers of homogeneous and spatially independent random simplicial complexes

Abstract

The Linial-Meshulam complex model is a natural higher dimensional analog of the Erdős-Rényi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial-Meshulam complexes with an appropriate scaling of the underlying parameter. The present article aims to extend that result to more general random simplicial complex models. We introduce a class of homogeneous and spatially independent random simplicial complexes, including the Linial-Meshulam complex model and the random clique complex model as special cases, and we study the asymptotic behavior of their Betti numbers. Moreover, we obtain the convergence of the empirical spectral distributions of their Laplacians. A key element in the argument is the local weak convergence of simplicial complexes. Inspired by the work of Linial and Peled, we establish the local weak limit theorem for homogeneous and spatially independent random simplicial complexes.

Construction of simplicial complexes with prescribed degree-size sequences

“…The decay of the average number of cycles β1 β 1 β 1 when average degree d is increased is reminiscent of the law of large numbers for Betti numbers in random simplicial complexes, such as the Linial-Meshulam complex [45] and the random clique complex [46]. We note that in these studies, the limiting behavior of Betti numbers is discussed in the context of increasing facet density, where the decay is driven by filling the k-dimensional cycles with (k + 1)-simplices.…”

Central Limit Theorem for Euclidean Minimal Spanning Acycles

“…As in the higher dimensional case, we must choose ε > 0 small enough so that the combinatorial structure of the complex changes in a predictable way. Therefore, as in (24), we can choose ε < r 40 , and set A(q i ) = B ε (q i ). Let a cone at a point x in the direction − → v of angle α be denoted by Cone(x, − → v , α).…”

“…Though the concept of spanning acycles was introduced by Kalai [23] in 1983, minimal spanning acycles have only received attention in recent years [16] and especially in the context of random simplicial complexes [19,40,18,24].…”

“…We extend this to X as in statement of the lemma by a perturbation argument for ε > 0 small enough. For d ≥ 3, choose ε > 0 such that (24) 2(d + 2)ε < r 4 , Now let X be as in the statement of the lemma and assume that Q := X ∩ B 3r (0) = {p 1 , . .…”

Large deviations for subcomplex counts and Betti numbers in multi-parameter simplicial complexes

“…This includes the existence of a dominating dimension and central limit theorems for the Euler characteristic; see e.g. Kahle and Meckes (2013); Thoppe et al (2016); Fowler (2019); Kanazawa (2022). Functional limit theorems for a dynamic version of the multiparameter model have been established in Owada et al (2021).…”