On the Vanishing of Homology in Random \v{C}ech Complexes
Probability
2016-03-24 v2 Algebraic Topology
Combinatorics
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.
Cite
@article{arxiv.1507.06945,
title = {On the Vanishing of Homology in Random \v{C}ech Complexes},
author = {Omer Bobrowski and Shmuel Weinberger},
journal= {arXiv preprint arXiv:1507.06945},
year = {2016}
}