  Given a family $\mathcal G$ of graphs on a common vertex set $X$, we say that
$\mathcal G$ is rainbow connected if for every vertex pair $u,v \in X$, there
exists a path from $u$ to $v$ that uses at most one edge from each graph in
$\mathcal G$. We consider the case that $\mathcal G$ contains $s$ graphs, each
sampled randomly from $G(n,p)$, with $n = |X|$ and $p = \frac{c \log n}{sn}$,
where $c > 1$ is a constant. We show that when $s$ is sufficiently large,
$\mathcal G$ is a.a.s. rainbow connected, and when $s$ is sufficiently small,
$\mathcal G$ is a.a.s. not rainbow connected. We also calculate a threshold of
$s$ for the rainbow connectivity of $\mathcal G$, and we show that this
threshold is concentrated on at most three values, which are larger than the
diameter of the union of $\mathcal G$ by about $\frac{\log n}{(\log \log
n)^2}$.
  The same results also hold in a more traditional random rainbow setting,
where we take a random graph $G\in G(n,p)$ with $p=\frac{c \log n}{n}$ ($c>1$)
and color each edge of $G$ with a color chosen uniformly at random from the set
$[s]$ of $s$ colors.
