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We consider an Euclidean supersymmetric field theory in $Z^3$ given by asupersymmetric $\Phi^4$ perturbation of an underlying massless Gaussian measureon scalar bosonic and Grassmann fields with covariance the Green's function ofa (stable) L\'evy random walk in $Z^3$. The Green's function depends on theL\'evy-Khintchine parameter $\alpha={3+\epsilon\over 2}$ with $0<\alpha<2$. For$\alpha ={3\over 2}$ the $\Phi^{4}$ interaction is marginal. We prove for$\alpha-{3\over 2}={\epsilon\over 2}>0$ sufficiently small and initialparameters held in an appropriate domain the existence of a globalrenormalization group trajectory uniformly bounded on all renormalization groupscales and therefore on lattices which become arbitrarily fine. At the sametime we establish the existence of the critical (stable) manifold. Theinteractions are uniformly bounded away from zero on all scales and thereforewe are constructing a non-Gaussian supersymmetric field theory on all scales.The interest of this theory comes from the easily established fact that theGreen's function of a (weakly) self-avoiding L\'evy walk in $Z^3$ is a secondmoment (two point correlation function) of the supersymmetric measure governingthis model. The control of the renormalization group trajectory is apreparation for the study of the asymptotics of this Green's function. Therigorous control of the critical renormalization group trajectory is apreparation for the study of the critical exponents of the (weakly)self-avoiding L\'evy walk in $Z^3$.

The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3