Homological Connectivity in Random \v{C}ech Complexes

Understand

We study the homology of random \v{C}ech complexes generated by a homogeneous Poisson process.

• We focus on 'homological connectivity' - the stage where the random complex is dense enough, so that its homology "stabilizes" and becomes isomorphic to that of the underlying topological space.
• Our results form a comprehensive high-dimensional analogue of well-known phenomena related to connectivity in the Erd\H{o}s-R\'enyi graph and random geometric graphs.
• We first prove that there is a sharp phase transition describing homological connectivity.