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The global renormalization group trajectory in a critical supersymmetric field theory on the lattice \(\mathbb Z^{3}\). (English) Zbl 1161.82310
Summary: We consider an Euclidean supersymmetric field theory in \(\mathbb Z^{3}\) given by a supersymmetric \(\Phi ^{4}\) perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green’s function of a (stable) Lévy random walk in \(\mathbb Z^{3}\). The Green’s function depends on the Lévy-Khintchine parameter \(\alpha ={3+\varepsilon \over 2}\) with \(0<\alpha <2\). For \(\alpha ={3\over 2}\) the \(\Phi ^{4}\) interaction is marginal. We prove for \(\alpha -{3\over 2}={\varepsilon \over 2}>0\) sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green’s function of a (weakly) self-avoiding Lévy walk in \(\mathbb Z^{3}\) is a second moment (two point correlation function) of the supersymmetric measure governing this model. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in \(\mathbb Z^{3}\).

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B28 Renormalization group methods in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
81T60 Supersymmetric field theories in quantum mechanics
60G51 Processes with independent increments; Lévy processes
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