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

Abstract

Rigorous upper bounds to the correlation functions and from there to the susceptibility and magnetization of spin-\textonehalf{} 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 ${T}_{0}$ which is thus a bound for the critical temperature ${T}_{c}$. It is hence proved that the mean-field and Bethe approximations yield upper bounds for ${T}_{c}$. Stronger bounds are presented; specifically, for $d$-dimensional isotropic hypercubical lattices, it is shown that $\frac{k{T}_{c}}{2dJ}\ensuremath{\le}1\ensuremath{-}(\frac{1}{2d})\ensuremath{-}(\frac{1}{3{d}^{2}})+O(\frac{1}{{d}^{3}})$ as $d\ensuremath{\rightarrow}\ensuremath{\infty}$. For anisotropic hypercubical lattices with interactions ${J}_{i}$ ($i=1, \ensuremath{\cdots}d$), the equality $\frac{k{T}_{c}}{{J}_{1}}=2{[\mathrm{ln}{\ensuremath{\eta}}^{\ensuremath{-}1}\ensuremath{-}\mathrm{lnln}{\ensuremath{\eta}}^{\ensuremath{-}1}+O(1)]}^{\ensuremath{-}1}$ is proved for all $d\ensuremath{\ge}2$ in the limit $\ensuremath{\eta}=\frac{({J}_{2}+{J}_{3}+\ensuremath{\cdots}+{J}_{d})}{{J}_{1}}\ensuremath{\rightarrow}0$.