Reconstruction thresholds on regular trees

Abstract

We consider a branching random walk with binary state space and index set $T^k$, the infinite rooted tree in which each node has k children (also known as the model of "broadcasting on a tree"). 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 2x2 transition matrix P. We say that "reconstruction is possible" if the values at the d'th level of the tree contain non-vanishing information about the value at the root as $d\to\infty$. Adapting 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 $p_{11}=0$. 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 $\lambda=1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any k.