Crumpling self-avoiding surfaces

Self-avoiding plaquette surfaces with a folding or bending fugacity are believed to undergo a “crumpling” transition from a flaccid phase with branched polymer characteristics (corresponding to surfaces with a high degree of folding), to a “smooth” phase (corresponding to surfaces faith a low degree of folding). I develop rigorous techniques in order to bound the free energy of this model. In particular, the limiting free energy is proven to be positive for all positive values of the folding fugacity. In addition, the existence of a nonanalyticity in the limiting free energy of a (nontrivial) subclass of surfaces is proven. This implies the existence of a phase transition in this model, which I conjecture to be from a “flaccid” to a “smooth” phase.