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Critical \((\Phi^4)_{3,\epsilon}\). (English) Zbl 1053.81065
Summary: The Euclidean \((\phi^4)_{3,\epsilon}\) model in \(\mathbb R^3\) corresponds to a perturbation by a \(\phi^4\) interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter \(\varepsilon\) in the range \(0\leq\varepsilon\leq1\). For \(\varepsilon=1\) one recovers the covariance of a massless scalar field in \(\mathbb R^3\). For \(\varepsilon =0\), \(\phi^4\) is a marginal interaction. For \(0\leq\varepsilon<1\) the covariance continues to be Osterwalder-Schrader and pointwise positive. We consider the infinite volume critical theory with a fixed ultraviolet cutoff at the unit length scale and we prove that for \(\varepsilon>0\), sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. We construct the stable critical manifold near this fixed point and prove that under Renormalization Group iterations the critical theories converge to the fixed point.

MSC:
81T08 Constructive quantum field theory
81T17 Renormalization group methods applied to problems in quantum field theory
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