Phase transition in random planar diagrams and RNA-type matching

We study the planar matching problem, defined by a symmetric random matrix with independent identically distributed entries, taking values zero and one. We show that the existence of a perfect planar matching structure is possible only above a certain critical density, p(c), of allowed contacts (i.e., of ones). Using a formulation of the problem in terms of Dyck paths and a matrix model of planar contact structures, we provide an analytical estimation for the value of the transition point, p(c), in the thermodynamic limit. This estimation is close to the critical value, p(c)≈0.379, obtained in numerical simulations based on an exact dynamical programming algorithm. We characterize the corresponding critical behavior of the model and discuss the relation of the perfect-imperfect matching transition to the known molten-glass transition in the context of random RNA secondary structure formation. In particular, we provide strong evidence supporting the conjecture that the molten-glass transition at T=0 occurs at p(c).