Reconstruction Thresholds on Regular Trees

We consider the model of broadcasting on a tree, with binary state space, on the infinite rooted tree k in which each node has k children. The root of the tree takes a random value 0 or 1, and then each node passes a value independently to each of its children according to a 2 2 transition matrix P. We say that reconstruction is possible if the values at the dth level of the tree contain non-vanishing information about the value at the root as d ∞. Extending a method of Brightwell and Winkler, we obtain new conditions under which reconstruction is impossible, both in the general case and in the special case p11 0. The latter case is closely related to the hard-core model from statistical physics; a corollary of our results is that, for the hard-core model on the k 1 -regular tree with activity λ 1, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any k 2.