Title: RG analysis for a weakly self-avoiding Levy walk in the lattice
Z^3.
Abstract:
The Green's function of a (weakly) self-avoiding L\'evy walk in a
large but finite cube in Z^3 can be expressed as the two point
correlation function in a supersymmetric field theory in this
cube. The underlying super-measure has been rigorously analyzed for a
particular class of walks by methods of the Wilson RG based on finite
range multi-scale expansions. We prove in an appropriate Banach space
setup that, for initial parameters of the interaction measuring
self-repulsion held in an appropriate domain, there exists a global
renormalization group trajectory uniformly bounded on all
renormalization group scales and therefore on lattices which become
arbitrarily fine. We establish the existence of the critical (stable)
manifold. Moreover we prove that interactions are uniformly bounded
away from zero on all scales. This is a step in a program to study
the critical exponents of of weakly self-avoiding Levy walks.
Based on joint work with Benedetto Scoppola published in
J.Stat Phys (2008) 133:921-1011.