  Let $G=G(n,m)$ be a random graph whose average degree $d=2m/n$ is below the
$k$-colorability threshold. If we sample a $k$-coloring $\sigma$ of $G$
uniformly at random, what can we say about the correlations between the colors
assigned to vertices that are far apart? According to a prediction from
statistical physics, for average degrees below the so-called {\em condensation
threshold} $d_c(k)$, the colors assigned to far away vertices are
asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences
2007]. We prove this conjecture for $k$ exceeding a certain constant $k_0$.
More generally, we investigate the joint distribution of the $k$-colorings that
$\sigma$ induces locally on the bounded-depth neighborhoods of any fixed number
of vertices. In addition, we point out an implication on the reconstruction
problem.
