N=2 Supersymmetry, Painleve III and Exact Scaling Functions in 2D Polymers

We discuss in this paper various aspects of the off-critical $O(n)$ model in two dimensions. We find the ground-state energy conjectured by Zamolodchikov for the unitary minimal models, and extend the result to some non-unitary minimal cases. We apply our results to the discussion of scaling functions for polymers on a cylinder. We show, using the underlying N=2 supersymmetry, that the scaling function for one non-contractible polymer loop around the cylinder is simply related to the solution of the Painleve III differential equation. We also find the ground-state energy for a single polymer on the cylinder. We check these results by numerically simulating the polymer system. We also analyze numerically the flow to the dense polymer phase. We find there surprising results, with a $c_{\hbox{eff}}$ function that is not monotonous and seems to have a roaming behavior, getting very close to the values 81/70 and 7/10 between its UV and IR values of 1.