  The Linial-Meshulam complex model is a natural higher-dimensional analog of
the Erd\H{o}s-R\'enyi 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 paper 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.
