Inside the clustering window 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. Previous work has established a clustering threshold, cr for this model: if c = c ∗ r − ǫ for any constant ǫ > 0 then with high probability all solutions form a well-connected cluster; whereas if c = cr + ǫ, then with high probability the solutions partition into wellconnected, well-separated clusters (with probability tending to 1 as n → ∞). 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 = cr + o(1), and prove that the connectivity parameters of the r-XORSAT clusters undergo a smooth transition around the clustering threshold.