Phase transition in random planar diagrams and RNA-type matching

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 2013. Vol. 88. P. 052117.

Lokhov A., Valba O. V., Tamm M., Nechaev S.

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 of allowed contacts, $p_{c}$. This problem has an important application for the prediction of the optimal folding of RNA-type polymers. Using an alternative 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, $p_{c}\approx 0.38$, obtained in numerical simulations based on an exact dynamic-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 $p_{c}$

Keywords: RNA secondary structure, graph matching, phase transitions, planar diagrams