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

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