Broadcasting on Trees and the Ising Model

• Published in: The Annals of applied probability 2000-05, Vol.10 (2), p.410-433
• Main Authors: Evans, William, Kenyon, Claire, Peres, Yuval, Schulman, Leonard J.
• Format: Article
• Language: English
• Subjects: 60K35; 68R99; 90B15; branching number; Children; electrical network; Ising model; Leaves; Markov chains; Mathematical theorems; noisy computation; Parsimony; percolation; Plant roots; Probabilities; Random variables; Tree; Vertices
• ISSN: 1050-5164; 2168-8737
• DOI: 10.1214/aoap/1019487349

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 $(1 - 2\varepsilon)^2 < p_c(T)$, but the reconstruction probability stays bounded away from 1/2 if the opposite inequality holds. Here pc(T) is the critical probability for percolation on T; in particular pc(T) = 1/b for the b + 1-regular tree. The asymptotic reconstruction problem is equivalent to purity of the "free boundary" Gibbs state for the Ising model on a tree. The special case of regular trees was solved in 1995 by Bleher, Ruiz and Zagrebnov; our extension to general trees depends on a coupling argument and on a reconstruction algorithm that weights the input bits by the electrical current flow from the root to the leaves.