A Non-trivial Fixed Point in a Three Dimensional Quantum Field Theory

International audienceWe report on the rigorous construction of an analogue of the Wilson-Fisher fixed point in three dimensions. The model corresponds to a perturbation by a $\phi^4$ interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter $\epsilon$ in the range $0\le \epsilon \le 1$. For $\epsilon =1$ one recovers the covariance of a massless scalar field in ${\bf R}^3$. For $\epsilon =0$, $\phi^{4}$ is a marginal interaction. For $0\le \epsilon 0$, 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