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

Abstract: The Erdős–Rényi graph model has been extensively studied since the 1960s as a typical random graph model. Recently, the study of random simplicial complexes has drawn attention as a higher-dimensional generalization of random graphs. In this talk we introduce a class of homogeneous and spatially independent random simplicial complexes, and discuss the asymptotic behavior of their Betti numbers. This result extends the law of large numbers for Betti numbers of Linial–Meshulam complexes, obtained in an earlier study by Linial and Peled. Time permitting, we will also discuss 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.

computational geometryalgebraic topologycombinatoricsgeometric topologyprobability

Audience: advanced learners

Asia Pacific Seminar on Applied Topology and Geometry