The Statistical Mechanics of Vesicles
In my first talk I will 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. Physically, this model is of interest as its phase diagram contains a point at which tricritical scaling is observed. By adding the constraint of partial directedness, one gets combinatorial models related to the partitioning of integers. These models are solvable in the sense that an explicit expression for the generating function can be given. The derivation of this generating function involves a non-linear functional equation, and the solution is given in terms of alternating $q$-series.