Critical Temperatures of Anisotropic Ising Lattices. II. General Upper Bounds

Rigorous upper bounds to the correlation functions and from there to the susceptibility and magnetization of spin-{} Ising models with general ferromagnetic interactions are obtained in terms of the generating functions for self-avoiding random walks on the corresponding lattice. These results are used to show that the spontaneous magnetization vanishes and the initial susceptibility is finite above a certain temperature T₀ which is thus a bound for the critical temperature Tc. It is hence proved that the mean-field and Bethe approximations yield upper bounds for Tc. Stronger bounds are presented; specifically, for d-dimensional isotropic hypercubical lattices, it is shown that kTc2dJ≤1-(1/2d)-(13d²)+O(1d³) as d→∞. For anisotropic hypercubical lattices with interactions Jᵢ (i=1, ⋯d), the equality kTcJ₁=2[lnη^-1-lnlnη^-1+O(1)]^-1 is proved for all d≥2 in the limit η=(J₂+J₃+⋯+Jd)J₁→0.