P.K. Mitter, B. Scoppola
The Global Renormalization Group Trajectory in a Critical 
Supersymmetric Field theory in Z^3 
(257K, TeX with attached macro)

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.