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2009 A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees
Christof Külske, Marco Formentin
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Electron. Commun. Probab. 14: 587-596 (2009). DOI: 10.1214/ECP.v14-1516


We give a criterion for the non-reconstructability of tree-indexed $q$-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix $M$. Non-reconstruction holds if $c(M)$ times the expected number of offspring on the Galton-Watson tree is smaller than 1. Here $c(M)$ is an explicit function, which is convex over the set of $M$'s with a given invariant distribution, that is defined in terms of a $(q-1)$-dimensional variational problem over symmetric entropies. This result is equivalent to proving the extremality of the free boundary condition Gibbs measure within the corresponding Gibbs-simplex. Our theorem holds for possibly non-reversible $M$ and its proof is based on a general recursion formula for expectations of a symmetrized relative entropy function, which invites their use as a Lyapunov function. In the case of the Potts model, the present theorem reproduces earlier results of the authors, with a simplified proof, in the case of the symmetric Ising model (where the argument becomes similar to the approach of Pemantle and Peres) the method produces the correct reconstruction threshold), in the case of the (strongly) asymmetric Ising model where the Kesten-Stigum bound is known to be not sharp the method provides improved numerical bounds.


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Christof Külske. Marco Formentin. "A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees." Electron. Commun. Probab. 14 587 - 596, 2009.


Accepted: 24 December 2009; Published: 2009
First available in Project Euclid: 6 June 2016

zbMATH: 1193.60117
MathSciNet: MR2576754
Digital Object Identifier: 10.1214/ECP.v14-1516

Primary: 60K35
Secondary: 82B20 , 82B44

Keywords: Broadcasting on trees , free boundary condition , Galton-Watson tree , Gibbs measures , Random tree , reconstruction problem

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