  We study the planar matching problem, defined by a symmetric random matrix
with independent identically distributed entries, taking values 0 and 1. 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 '1').
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, $p_{c} \approx 0.379$, 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's formation. In particular, we
provide strong evidence supporting the conjecture that the molten-glass
transition at T=0 occurs at $p_{c}$.
