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 T₂. It is hence proved that the mean-field and Bethe approximations yield upper bounds for T₂. Stronger bounds are presented; specifically, for d-dimensional isotropic hypercubical lattices, it is shown that kT₂2dJ1- (12d) - (13{d^2}) +O (1{d^3}) as d. For anisotropic hypercubical lattices with interactions J₈ (i=1, ), the equality kT₂{J₁}=2ln{^-1-lnln^-1+O (1) }^-1 is proved for all d2 in the limit = (J₂+{J₃++J₃) }{J₁}0.