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 α 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 α=k2k−1(lnk+lnlnk+γd+o(1)), c=q(lnq+lnlnq+γd+o(1)), where γd is the same constant for both models. We
characterize γd via a functional equation, solve the latter
numerically to estimate γd≈0.871, and obtain an analytic
lowerbound γd≥1+ln(2(2−1))≈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 γr=1.Comment: 35 pages, 8 figure