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→ ∞, the first homology group over 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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- Hitting time
- Homology Shadow
- Random simplicial complexes