We set up and solve the Bethe approximation for a spin-glass with finite-range interactions. We obtain the finite-z analog of the Thouless-Anderson-Palmer equations and the free-energy functional at all T. In the configuration-averaged theory the entropy S̄(T) and all other properties are well behaved on a Cayley tree. For real lattices S̄(0)=0, but the specific heat is negative near $T=0$ for all but small values of z.
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Bowman et al. (1982) studied this question.
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