  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 $\alpha = \frac{2^{k-1}}{k} (\ln k + \ln
\ln k + \gamma_{\rm d} + o(1))$, $c= q (\ln q + \ln \ln q + \gamma_{\rm d}+
o(1))$, where $\gamma_{\rm d}$ is the same constant for both models. We
characterize $\gamma_{\rm d}$ via a functional equation, solve the latter
numerically to estimate $\gamma_{\rm d} \approx 0.871$, and obtain an analytic
lowerbound $\gamma_{\rm d} \ge 1 + \ln (2 (\sqrt{2}-1)) \approx 0.812$. Our
analysis unveils a subtle interplay of the clustering transition with the
rigidity (naive reconstruction) threshold that occurs on the same asymptotic
scale at $\gamma_{\rm r}=1$.
