Abstract
The random 2-dimensional simplicial complex process starts with a complete graph on n vertices, and in every step a new 2-dimensional face, chosen uniformly at random, is added. We prove that with probability tending to 1 as \(n\rightarrow \infty \), the first homology group over \(\mathbb {Z}\) vanishes at the very moment when all the edges are covered by triangular faces.
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Acknowledgements
This work was carried out when TŁ visited the Institute for Mathematical Research (FIM) of ETH Zürich. He would like to thank FIM for the hospitality and for creating a stimulating research environment. TŁ partially supported by NCN Grant 2012/06/A/ST1/00261. YP is grateful to the Azrieli foundation for the award of an Azrieli fellowship.
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Łuczak, T., Peled, Y. Integral Homology of Random Simplicial Complexes. Discrete Comput Geom 59, 131–142 (2018). https://doi.org/10.1007/s00454-017-9938-z
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DOI: https://doi.org/10.1007/s00454-017-9938-z