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On the purity of the free boundary condition Potts measure on random trees

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  • Formentin, Marco
  • Külske, Christof

Abstract

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.

Suggested Citation

  • Formentin, Marco & Külske, Christof, 2009. "On the purity of the free boundary condition Potts measure on random trees," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2992-3005, September.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:9:p:2992-3005
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    Cited by:

    1. Bissacot, Rodrigo & Endo, Eric Ossami & van Enter, Aernout C.D., 2017. "Stability of the phase transition of critical-field Ising model on Cayley trees under inhomogeneous external fields," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4126-4138.

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