Abstract
In this paper, we consider the λ-model for an arbitrary-order Cayley tree that has a disordered phase. Such a phase corresponds to a splitting Gibbs measure with free boundary conditions. In communication theory, such a measure appears naturally, and its extremality is related to the solvability of the non-reconstruction problem. In general, the disordered phase is not extreme; hence, it is natural to find a condition for their extremality. In the present paper, we present certain conditions for the extremality of the disordered phase of the λ-model.
Highlights
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The Potts models became one of the important models in statistical mechanics. It has been known [16] for a long time that, at sufficiently low temperatures, such a q-state Potts model on the Cayley tree has at least q + 1 translation-invariant Gibbs measures which are tree-indexed Markov chains
We recall that an splitting Gibbs measure (SGM) is called the disordered phase of the λ-model if u1 = 1, u2 = 1 is a solution of (6)
Summary
For models of statistical physics on trees, the problem is related to the extremality of the disordered Gibbs measure [6,7]. This problem is related to the solvability of the reconstruction problem of Markov fields on trees (see [8]). The Potts models became one of the important models in statistical mechanics It has been known [16] for a long time that, at sufficiently low temperatures, such a q-state Potts model on the Cayley tree has at least q + 1 translation-invariant Gibbs measures which are tree-indexed Markov chains. We are going to investigate extremity and non-extremity of translation-invariant Gibbs measures of the λ-model.
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