  Let $G_n$ be the binomial random graph $G(n,p=c/n)$ in the sparse regime,
which as is well-known undergoes a phase transition at $c=1$.
  Lynch (Random Structures Algorithms, 1992) showed that for every first order
sentence $\phi$, the limiting probability that $G_n$ satisfies $\phi$ as
$n\to\infty$ exists, and moreover it is an analytic function of $c$. In this
paper we consider the closure $\overline{L_c}$ in $[0,1]$ of the set $L_c$ of
all limiting probabilities of first order sentences in $G_n$. We show that
there exists a critical value $c_0 \approx0.93$ such that $\overline{L_c}=
[0,1]$ when $c \ge c_0$, whereas $\overline{L_c}$ misses at least one
subinterval when $c<c_0$. We extend these results to random $d$-uniform sparse
hypergraphs, where the probability of a hyperedge is given by $p=c/n^{d-1}$.
