Extremality of the Disordered State for the Ising Model on General Trees

We develop a method to study extremality of the disordered state ℙ β for the Ising model on a general countable tree T. It is shown that the tail σ- field is ℙ β -trivial as soon as β is less than the spin glass critical inverse temperature $$ \beta _c^{SG} $$ , which is determined from the relation tanh $$ (\beta _c^{SG} ) = {1 \mathord{\left/ {\vphantom {1 {\sqrt {br(T)} }}} \right. \kern-\nulldelimiterspace} {\sqrt {br(T)}}} $$ . 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.

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