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A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees
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Christof Külske, University of Bochum
Marco Formentin, University of Bochum
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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.
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Full text: PDF
Pages: 587-596
Published on: December 24, 2009
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Electronic Communications in Probability. ISSN: 1083-589X |
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