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arXiv · 1507.06945

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

Omer Bobrowski·Shmuel Weinberger
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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.

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Omer Bobrowski, Shmuel Weinberger. 2015-07-24. On the Vanishing of Homology in Random \v{C}ech Complexes. https://arxiv.org/abs/1507.06945

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