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Perturbation about the mean field critical point
Authors:Jean Bricmont  Jean-Raymond Fontaine  Eugene Speer
Institution:(1) Department of Mathematics, Princeton University, 08544 Princeton, NJ, USA;(2) Department of Mathematics, Rutgers University, 08903 New Brunswick, NJ, USA;(3) Present address: Institut de Physique Theorique, Université de Louvain, Belgium
Abstract:We consider two models that are small perturbations of Gaussian or mean field models: the first one is a double well lambda/4phgr4sgr/2phgr2 perturbation of a massless Gaussian lattice field in the weak coupling limit (lambdararr0, sgr proportional to lambda). The other consists of a spin 1/2 Ising model with long-range Kac type interactions; the inverse range of the interaction, gamma, is the small parameter. The second model is related to the first one via a sine-Gordon transformation. The lattice Zopf d has dimensiondgE3.In both cases we derive an asymptotic estimate to first order (in lambda or gamma2) on the location of the critical point. Moreover, we prove bounds on the remainder of an expansion in lambda or gamma 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.Supported by NSF Grant # MCS 78-01885Supported by NSF Grant # PHY 78-15920
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