Inside the clustering threshold for random linear equations

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.