Perturbation about the mean field critical point

Abstract

We consider two models that are small perturbations of Gaussian or mean field models: the first one is a double well λ/4φ4 — σ/2φ2 perturbation of a massless Gaussian lattice field in the weak coupling limit (λ→0, σ proportional to λ). The other consists of a spin 1/2 Ising model with long-range Kac type interactions; the inverse range of the interaction, γ, is the small parameter. The second model is related to the first one via a sine-Gordon transformation. The lattice ℤd has dimensiond≧3.In both cases we derive an asymptotic estimate to first order (in λ or γ2) on the location of the critical point. Moreover, we prove bounds on the remainder of an expansion in λ or γ around the Gaussian or mean field critical points.The appendix, due to E. Speer, contains an extension of Weinberg's theorem on the divergence of Feynman graphs which is used in the proofs.