New phase transition in random planar diagrams and RNA-type matching

TL;DR

This paper investigates a phase transition in the planar matching problem, showing that perfect matchings only occur above a critical contact density, with implications for RNA secondary structure formation.

Contribution

It introduces an analytical estimation of the critical contact density for perfect matchings in random planar diagrams, linking it to RNA folding transitions.

Findings

Critical contact density estimated at p_c ≈ 0.379

Perfect matchings only exist above p_c

Transition related to RNA molten-glass transition

Abstract

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 $p_{c}$.