The Annals of Applied Probability

Broadcasting on trees and the Ising model

William Evans, Claire Kenyon, Yuval Peres, and Leonard J. Schulman

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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 $\varepsilon$, 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.

Article information

Source
Ann. Appl. Probab. Volume 10, Number 2 (2000), 410-433.

Dates
First available in Project Euclid: 22 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1019487349

Digital Object Identifier
doi:10.1214/aoap/1019487349

Mathematical Reviews number (MathSciNet)
MR1768240

Zentralblatt MATH identifier
1052.60076

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 90B15: Network models, stochastic 68R99: None of the above, but in this section

Keywords
Tree percolation Ising model branching number electrical network noisy computation

Citation

Evans, William; Kenyon, Claire; Peres, Yuval; Schulman, Leonard J. Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 (2000), no. 2, 410--433. doi:10.1214/aoap/1019487349. https://projecteuclid.org/euclid.aoap/1019487349.


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  • LRI, Universit´e de Paris-Sud Orsay France E-mail: Claire.Kenyon@lri.fr