The threshold function for vanishing of the top homology group of random $d$-complexes

Abstract

For positive integers $n$ and $d$, and the probability function $0\leq p(n)\leq 1$, we let $Y_{n,p,d}$ denote the probability space of all at most $d$-dimensional simplicial complexes on $n$ vertices, which contain the full $(d-1)$-dimensional skeleton, and whose $d$-simplices appear with probability $p(n)$. In this paper we determine the threshold function for vanishing of the top homology group in $Y_{n,p,d}$, for all $d\geq 1$.