  Denote by an $\ell$-component a connected $b$-uniform hypergraph with $k$
edges and $k(b-1) - \ell$ vertices. We prove that the expected number of
creations of $\ell$-component during a random hypergraph process tends to 1 as
$\ell$ and $b$ tend to $\infty$ with the total number of vertices $n$ such that
$\ell = o(\sqrt[3]{\frac{n}{b}})$. Under the same conditions, we also show that
the expected number of vertices that ever belong to an $\ell$-component is
approximately $12^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}$. As an immediate
consequence, it follows that with high probability the largest $\ell$-component
during the process is of size $O((b-1)^{1/3} \ell^{1/3} n^{2/3})$. Our results
give insight about the size of giant components inside the phase transition of
random hypergraphs.
