  Vesicles, or closed fluctuating membranes, have been modeled in two
dimensions by self-avoiding polygons, weighted with respect to their perimeter
and enclosed area, with the simplest model given by area-weighted excursions.
These models generically show a tricritical phase transition between an
inflated and a crumpled phase, with a scaling function given by the logarithmic
derivative of the Airy function. Extending such a model, we find realizations
of multicritical points of arbitrary order, with the associated multivariate
scaling functions expressible in terms of generalized Airy integrals, as
previously conjectured by John Cardy. This work therefore adds to the small
list of models with a critical phase transition, for which exponents and the
associated scaling functions are explicitly known.
