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Description:
We develop a method to study extremality of the disordered state ℙβ for the Ising model on a general countable tree T. It is shown that the tail σ-field is ℙβ-trivial as soon as β is less than the spin glass critical inverse temperature βS Gc , which is determined from the relation tanh(βS Gc) = 1/√br(T). The method is based on the FK representation of ferromagnetic systems and recursive estimates on conditional expectations of the spin at the root. Similar estimates in the context of the bit reconstruction problem on general trees were originally obtained in [EKPS] using different methods.