The integer homology threshold in 𝑌_{𝑑}(𝑛,𝑝)

Abstract

We prove that in the d d -dimensional Linial–Meshulam stochastic process the ( d − 1 ) (d - 1) st homology group with integer coefficients vanishes exactly when the final maximal ( d − 1 ) (d - 1) -dimensional face is covered by a top-dimensional face. This generalizes the d = 2 d = 2 case proved recently by Łuczak and Peled [Discrete Comput. Geom. 59 (2018), pp. 131–142] and establishes that p = d log ⁡ n / n p =d \log n/n is the sharp threshold for homology with integer coefficients to vanish in Y d ( n , p ) Y_d(n, p) , answering a 2003 question of Linial and Meshulam [Combinatorica 26 (2006), pp. 475–487].