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May 2000 Broadcasting on trees and the Ising model
William Evans, Claire Kenyon, Yuval Peres, Leonard J. Schulman
Ann. Appl. Probab. 10(2): 410-433 (May 2000). 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 $\varepsilon$, and these errors occur independently. The goal is to reconstruct $R$ from the values which arrive at the $n$th 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 $n \to \infty$ if $(1 - 2\varepsilon)^2 < p_c(T)$, but the reconstruction probability stays bounded away from ½ if the opposite inequality holds. Here $p_c(T)$ is the critical probability for percolation on $T$; in particular $p_c(T) = 1/b$ 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 algorighm that weights the input bits by the electrical current flow from the root to the leaves.


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William Evans. Claire Kenyon. Yuval Peres. Leonard J. Schulman. "Broadcasting on trees and the Ising model." Ann. Appl. Probab. 10 (2) 410 - 433, May 2000.


Published: May 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1052.60076
MathSciNet: MR1768240
Digital Object Identifier: 10.1214/aoap/1019487349

Primary: 60K35
Secondary: 68R99 , 90B15

Keywords: branching number , electrical network , Ising model , noisy computation , percolation , tree

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.10 • No. 2 • May 2000
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