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

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