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The critical Ising model on trees, concave recursions and nonlinear capacity. (English) Zbl 1197.60092

The paper deals with the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. The authors determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. Exact capacity criteria that govern behavior at critical temperatures was obtained. The authors derive necessary and sufficient condition under which there is a unique Gibbs measure for the ferromagnetic Ising model at the relevant critical temperature.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
31C45 Other generalizations (nonlinear potential theory, etc.)
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