  We study a random system of $cn$ linear equations over $n$ variables in
GF(2), where each equation contains exactly $r$ variables; this is equivalent
to $r$-XORSAT. \cite{ikkm,amxor} determined the clustering threshold, $c^*_r$:
if $c=c^*_r+\e$ for any constant $\e>0$, then \aas the solutions partition into
well-connected, well-separated {\em clusters} (with probability tending to 1 as
$n\rightarrow\infty$). This is part of a general clustering phenomenon which is
hypothesized to arise in most of the commonly studied models of random
constraint satisfaction problems, via sophisticated but mostly non-rigorous
techniques from statistical physics. We extend that study to the range
$c=c^*_r+o(1)$, showing that if $c=c^*_r+n^{-\d}, \d>0$, then the connectivity
parameter of each $r$-XORSAT cluster is $n^{\Theta(\d)}$, as compared to
$O(\log n)$ when $c=c^*_r+\e$. This means that one can move between any two
solutions in the same cluster via a sequence of solutions where consecutive
solutions differ on at most $n^{\Theta(\d)}$ variables; this is tight up to the
implicit constant. In contrast, moving to a solution in another cluster
requires that some pair of consecutive solutions differ in at least
$n^{1-O(\d)}$ variables.
  Along the way, we prove that in a random $r$-uniform hypergraph with
edge-density $n^{-\d}$ above the $k$-core threshold, \aas every vertex not in
the $k$-core can be removed by a sequence of $n^{\Theta(\d)}$ vertex-deletions
in which the deleted vertex has degree less than $k$; again, this is tight up
to the implicit constant.
