  We consider connected components in $k$-uniform hypergraphs for the following
notion of connectedness: given integers $k\ge 2$ and $1\le j \le k-1$, two
$j$-sets (of vertices) lie in the same $j$-component if there is a sequence of
edges from one to the other such that consecutive edges intersect in at least
$j$ vertices.
  We prove that certain collections of $j$-sets constructed during a
breadth-first search process on $j$-components in a random $k$-uniform
hypergraph are reasonably regularly distributed with high probability. We use
this property to provide a short proof of the asymptotic size of the giant
$j$-component shortly after it appears.
