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Renormalization in Quantum Field Theory, Statistical Mechanics, and Condensed Matter, Vienne : Autriche (2007)
The global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3
P. K. Mitter1

We consider an Euclidean supersymmetric field theory in $\math{Z}^{3}$ 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 $\math{Z}^{3}$. The Green's function depends on the Lévy-Khintchine parameter $\a={3+\e\over 2}$ with $0<\a<2$. For $\a ={3\over 2}$ the $\Phi^{4}$ interaction is marginal. We prove for $\a-{3\over 2}={\e\over 2}>0$ 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 $\math{Z}^{3}$ 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 $\math{Z}^{3}$.
1 :  LPTA - Laboratoire de Physique Théorique et Astroparticules
Physique/Physique mathématique

Mathématiques/Physique mathématique

Physique/Physique des Hautes Energies - Théorie