Reconstruction for Models on Random Graphs

Consider a collection of random variables attached to the vertices of a graph. The reconstruction problem requires to estimate one of them given far away´ observations. Several theoretical results (and simple algorithms) are available when (heir joint probal)ility distribution is Markov with respect to a tree. In this paper we consider the case of sequences of random graphs that converge locally to trees. In particular, we develop a sufficient condition for the tree and graph reconstruction problem to coincide. We apply such condition to colorings of random graphs. Further, we characterize the behavior of I´sing models on such graphs, both with attractive and random interactions (respectively, ferromagnetic´ and ´spin glass´).