  Motivated by the theory of spin-glasses in physics, we study the so-called
reconstruction problem for the related distributions on the tree, and on the
sparse random graph $G(n,d/n)$.
  Both cases, reduce naturally to studying broadcasting models on the tree,
where each edge has its own broadcasting matrix, and this matrix is drawn
independently from a predefined distribution. In this context, we study the
effect of the configuration at the root to that of the vertices at distance
$h$, as $h\to\infty$.
  We establish the reconstruction threshold for the cases where the
broadcasting matrices give rise to symmetric, 2-spin Gibbs distributions. This
threshold seems to be a natural extension of the well-known Kesten-Stigum bound
which arises in the classic version of the reconstruction problem.
  Our results imply, as a special case, the reconstruction threshold for the
well-known Edward-Anderson model of spin-glasses on the tree.
  Also, we extend our analysis to the setting of the Galton-Watson tree, and
the random graph $G(n,d/n)$, where we establish the corresponding
thresholds.Interestingly, for the Edward-Anderson model on the random graph, we
show that the replica symmetry breaking phase transition, established in
[Guerra and Toninelli:2004], coincides with the reconstruction threshold.
  Compared to the classical Gibbs distributions, the spin-glasses have a lot of
unique features. In that respect, their study calls for new ideas, e.g., we
introduce novel estimators for the reconstruction problem. Furthermore, note
that the main technical challenge in the analysis is the presence of (too) many
levels of randomness. We manage to circumvent this problem by utilising
recently proposed tools coming from the analysis of Markov chains.
