Abstract
Let \(Y \sim Y_d(n,p)\) denote the Bernoulli random d-dimensional simplicial complex. We answer a question of Linial and Meshulam from 2003, showing that the threshold for vanishing of homology \(H_{d-1}(Y; \mathbb {Z})\) is less than \(40d (d+1) \log n / n\). This bound is tight, up to a constant factor which depends on d.
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Acknowledgements
The authors thank Nati Linial and Roy Meshulam for many helpful and encouraging conversations. C.H. gratefully acknowledges support from NSF Grant DMS-1308645 and NSA Grant H98230-13-1-0827. M.K. gratefully acknowledges support from the Alfred P. Sloan Foundation, from DARPA Grant N66001-12-1-4226, and from NSF Grant CCF-1017182. E.P. gratefully acknowledges support from NSF Grant DMS-0847661.
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Hoffman, C., Kahle, M. & Paquette, E. The Threshold for Integer Homology in Random d-Complexes. Discrete Comput Geom 57, 810–823 (2017). https://doi.org/10.1007/s00454-017-9863-1
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DOI: https://doi.org/10.1007/s00454-017-9863-1