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 $\gamma >2\mathrm{}$, the approximation predicts, among other things, a phase transition in zero field at $T_c=2J {\left\{\ln \right[\frac{\gamma }{(\gamma -2)}\left]\right\}}^{-1\mathrm{}}$, 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.