Scaling behavior of tethered crumpled manifolds with inner dimension close to D=2: Resumming the perturbation theory

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 the summation of the perturbation expansion in the strength g0 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 to interpolate to the well studied case of polymers (D=1). We give results to fourth order in 2−D, where the dependence on g0 is again summed exactly. As an application, we discuss plaquette density functions, and propose a Monte Carlo experiment to test our results. These methods shed light on the more complex problem of self-avoiding manifolds.