Spaces of completions of elementary theories and convergence laws for random hypergraphs

Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p:\mathbb{N}\to[0,1]$. We prove that, for all logarithmo-exponential $p\ll n^{-d+\epsilon}$, the probabilities of all elementary properties of hypergraphs converge, with particular emphasis in the ranges $p(n)\sim C/n^d$ and $p(n) \sim C\log(n)/n^d$. The exposition is unified by constructing, for each such function $p$, the topological space of all completions of its almost sure theory. This space turns out to be compact, metrizable and totally disconnected, but further properties depend on the range of $p$. The convergence of the probabilities of elementary properties is associated with a borelian probability measure on the space.