The Threshold for Integer Homology in Random d-Complexes

Abstract

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(YźZ) 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.