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Random Constraint Satisfaction Problems exhibit several phase transitionswhen their density of constraints is varied. One of these threshold phenomena,known as the clustering or dynamic transition, corresponds to a transition foran information theoretic problem called tree reconstruction. In this article westudy this threshold for two CSPs, namely the bicoloring of $k$-uniformhypergraphs with a density $\alpha$ of constraints, and the $q$-coloring ofrandom graphs with average degree $c$. We show that in the large $k,q$ limitthe clustering transition occurs for $\alpha = \frac{2^{k-1}}{k} (\ln k + \ln\ln k + \gamma_{\rm d} + o(1))$, $c= q (\ln q + \ln \ln q + \gamma_{\rm d}+o(1))$, where $\gamma_{\rm d}$ is the same constant for both models. Wecharacterize $\gamma_{\rm d}$ via a functional equation, solve the latternumerically to estimate $\gamma_{\rm d} \approx 0.871$, and obtain an analyticlowerbound $\gamma_{\rm d} \ge 1 + \ln (2 (\sqrt{2}-1)) \approx 0.812$. Ouranalysis unveils a subtle interplay of the clustering transition with therigidity (naive reconstruction) threshold that occurs on the same asymptoticscale at $\gamma_{\rm r}=1$.

The asymptotics of the clustering transition for random constraint satisfaction problems