Random graph products of finite groups are rational duality groups

Given a Bernoulli random graph $G \sim G(n,p)$, we determine various facts about the cohomology of graph products of groups for the graph $G$. In particular, the random graph product of a sequence of finite groups is a rational duality group with probability tending to $1$ as $n \to \infty$. This includes random right angled Coxeter groups as a special case.