  The field theory of self-avoiding tethered membranes still poses major
challenges. In this article, we report progress on the toy-model of a manifold
repelled by a single point. Our approach allows to sum the perturbation
expansion in the strength g_0 of the interaction exactly in the limit of
internal dimension D -> 2, yielding an analytic solution for the
strong-coupling limit. This analytic solution is the starting point for an
expansion in 2-D, which aims at connecting to the well studied case of polymers
(D=1). We give results to fourth order in 2-D, where the dependence on g_0 is
again summed exactly. As an application, we discuss plaquette density
functions, and propose a Monte-Carlo experiment to test our results. These
methods should also allow to shed light on the more complex problem of
self-avoiding manifolds.
