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

TL;DR

This paper analyzes the homology of random Čech complexes over a Poisson process on a torus, revealing two main phase transitions related to connectivity and homology computation, with implications for understanding topological features at different scales.

Contribution

It provides a detailed analysis of phase transitions in the homology of random Čech complexes, including the first rigorous computation of these phenomena in this setting.

Findings

Identification of two main phase transitions in homology

Homology becomes fully computable at the second transition

Finer scale measurements suggest further homology separation

Abstract

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.