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

TL;DR

This paper proves the existence of a global renormalization group trajectory for a supersymmetric  field theory on a lattice, establishing a non-Gaussian critical model relevant for analyzing Le9vy walks in three dimensions.

Contribution

It demonstrates the construction of a non-Gaussian supersymmetric field theory with a controlled RG flow on all scales, including the critical manifold, for a lattice  model with Le9vy walk covariance.

Findings

Existence of a global RG trajectory bounded on all scales.

Construction of a non-Gaussian supersymmetric field theory.

Preparation for analyzing critical exponents of Le9vy walks.

Abstract

We consider an Euclidean supersymmetric field theory in $Z^3$ 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) L\'evy random walk in $Z^3$. The Green's function depends on the L\'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 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 L\'evy walk in $Z^3$ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The control of the renormalization group trajectory is a preparation for the study of the asymptotics of this Green's function. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding L\'evy walk in $Z^3$.