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.

N. Haug, T. Prellberg, Multicritical Scaling in a Lattice Model of Vesicles, (in review)