Journal
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
Volume 119, Issue 9, Pages 2992-3005Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.spa.2009.03.008
Keywords
Potts model; Gibbs measures; Random tree; Reconstruction problem; Free boundary condition
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We consider the free boundary condition Gibbs measure of the Potts model on a random tree. We provide an explicit temperature interval below the ferromagnetic transition temperature for which this measure is extremal, improving older bounds of Mossel and Peres. In information theoretic language extremality of the Gibbs measure corresponds to non-reconstructability for symmetric q-ary channels. The bounds for the corresponding threshold value of the inverse temperature are optimal for the Ising model and differ from the Kesten Stigum bound by only 1.50% in the case q = 3 and 3.65% for q = 4, independently of d. Our proof uses an iteration of random boundary entropies from the outside of the tree to the inside, along with a symmetrization argument. (C) 2009 Elsevier B.V. All rights reserved.
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