Cayley trees, the Ising problem, and the thermodynamic limit

Proofs have been given that the Bethe-Peierls approximation solves exactly the Ising problem on a Cayley tree. For a tree with coordination number >2, the approximation predicts, among other things, a phase transition in zero field at T₂=2J {ln{ (-2) }}^-1, with a discontinuity in the specific heat. On the other hand, the partition function in zero field can be calculated exactly and turns out to be analytic for all T. This paradox is analyzed and resolved. The transition occurring on a Cayley tree is found not to be of the type usually studied in thermodynamics.