On the Vanishing of Homology in Random \v{C}ech Complexes

Understand

We compute the homology of random \v{C}ech complexes over a homogeneous Poisson process on the d-dimensional torus, and show that there are, coarsely, two phase transitions.

• The first transition is analogous to the Erd\H{o}s-R\'enyi phase transition, where the \v{C}ech complex becomes connected.
• The second transition is where all the other homology groups are computed correctly (almost simultaneously).
• Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups.