  We consider a system of self-avoiding polygons interacting through a
potential that penalizes or rewards the number of mutual touchings and we
provide an exact computation of the critical curve separating a regime of long
polygons from a regime of localized polygons. Moreover, we prove the existence
of a sub-region of the phase diagram where the self-avoiding polygons are space
filling and we provide a non-trivial characterization of the regime where the
polygon length admits uniformly bounded exponential moments.
