The Asymptotics of the Clustering Transition for Random Constraint Satisfaction Problems

Abstract

Random constraint satisfaction problems exhibit several phase transitions when their density of constraints is varied. One of these threshold phenomena, known as the clustering or dynamic transition, corresponds to a transition for an information theoretic problem called tree reconstruction. In this article we study this threshold for two CSPs, namely the bicoloring of k-uniform hypergraphs with a density $$\alpha $$ of constraints, and the q-coloring of random graphs with average degree c. We show that in the large k, q limit the clustering transition occurs for $$\alpha = \frac{2^{k-1}}{k} (\ln k + \ln \ln k + \gamma _{\mathrm{d}} + o(1))$$ , $$c= q (\ln q + \ln \ln q + \gamma _{\mathrm{d}}+ o(1))$$ , where $$\gamma _{\mathrm{d}}$$ is the same constant for both models. We characterize $$\gamma _{\mathrm{d}}$$ via a functional equation, solve the latter numerically to estimate $$\gamma _{\mathrm{d}} \approx 0.871$$ , and obtain an analytic lowerbound $$\gamma _{\mathrm{d}} \ge 1 + \ln (2 (\sqrt{2}-1)) \approx 0.812$$ . Our analysis unveils a subtle interplay of the clustering transition with the rigidity (naive reconstruction) threshold that occurs on the same asymptotic scale at $$\gamma _{\mathrm{r}}=1$$ .