Frozen variables in random boolean constraint satisfaction problems

We determine the exact freezing threshold, r , for a family of models of random boolean constraint satisfaction problems, including NAE-SAT and hypergraph 2-colouring, when the constraint size is sufficiently large. If the constraint-density of a random CSP, F , in our family is greater than r then for almost every solution of F , a linear number of variables are frozen, meaning that their colours cannot be changed by a sequence of alterations in which we change o(n) variables at a time, always switching to another solution. If the constraintdensity is less than r , then almost every solution has o(n) frozen variables. Freezing is a key part of the clustering phenomenon that is hypothesized by non-rigorous techniques from statistical physics. The understanding of clustering has led to the development of advanced heuristics such as Survey Propogation. It has been suggested that the freezing threshold is a precise algorithmic barrier: that for densities below r the random CSPs can be solved using very simple algorithms, while for densities above r one requires more sophisticated techniques in order to deal with frozen clusters.