The Threshold for Integer Homology in Random d-Complexes

@article{Hoffman2017TheTF,
  title={The Threshold for Integer Homology in Random d-Complexes},
  author={Christopher Hoffman and Matthew Kahle and Elliot Paquette},
  journal={Discrete \& Computational Geometry},
  year={2017},
  volume={57},
  pages={810-823}
}
Let $$Y \sim Y_d(n,p)$$Y∼Yd(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 $$H_{d-1}(Y; \mathbb {Z})$$Hd-1(Y;Z) is less than $$40d (d+1) \log n / n$$40d(d+1)logn/n. This bound is tight, up to a constant factor which depends on d. 
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