Broadcasting on trees and the ising model

Abstract

Consider a process in which information is transmitted from a given root node on a noisy tree network T. We start with an unbiased random bit R at the root of the tree and send it down the edges of T. On every edge the bit can be reversed with probability ε, and these errors occur independently. The goal is to reconstruct R from the values which arrive at the nth level of the tree. This model has been studied in information theory, genetics and statistical mechanics. We bound the reconstruction probability from above, using the maximum flow on T viewed as a capacitated network, and from below using the electrical conductance of T. For general infinite trees, we establish a sharp threshold: the probability of correct reconstruction tends to 1/2 as n → ∞ if (1 - 2ε)2