Reconstruction Thresholds on Regular Trees

Abstract

We consider themodel of broadcasting on a tree, with binary state space, on theinfinite rooted tree $T^k$ in which each node has $k$ children. The root of the tree takesa random value $0$ or $1$, and then each node passes a value independently to each of its children according to a $2x2$ transition matrix $\mathbf{P}$. 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 $p_11=0$. 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 $λ =1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any $k ≥ 2$.