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A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees
2.	Creator	Author's name, affiliation, country	Christof Külske; University of Bochum; Germany
2.	Creator	Author's name, affiliation, country	Marco Formentin; University of Bochum; Germany
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Broadcasting on trees, Gibbs measures, random tree, Galton-Watson tree, recon- struction problem, free boundary condition
3.	Subject	Subject classification	60K35, 82B20, 82B44
4.	Description	Abstract	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2009-12-24
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1516
10.	Identifier	Digital Object Identifier	10.1214/ECP.v14-1516
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 14
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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