Combinatorial Enumeration of Two-Dimensional Vesicles

I discuss two-dimensional lattice models of closed fluctuating membranes, or vesicles. The underlying mathematical model is that of self--avoiding polygons and their enumeration by perimeter and area. By adding the constraint of partial directedness, one gets solvable models in the sense that an explicit expression for the generating function can be given in terms of alternating $q$--series. An asymptotic analysis leads to an explicit calculation of the scaling behaviour around the critical point in terms of the Airy function.