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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 tor-XORSAT. Previous work has established a clustering threshold, c^*_r for thismodel: if c=c_r^*-\epsilon for any constant \epsilon>0 then with highprobability all solutions form a well-connected cluster; whereas ifc=c^*_r+\epsilon, then with high probability the solutions partition intowell-connected, well-separated clusters (with probability tending to 1 as ngoes to infinity). This is part of a general clustering phenomenon which ishypothesized to arise in most of the commonly studied models of randomconstraint satisfaction problems, via sophisticated but mostly non-rigoroustechniques from statistical physics. We extend that study to the rangec=c^*_r+o(1), and prove that the connectivity parameters of the r-XORSATclusters undergo a smooth transition around the clustering threshold.

Inside the clustering window for random linear equations