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.