Phase transition in random planar diagrams and RNA-type matching

Area of Science:

• Statistical physics
• Combinatorics
• Computational biology

Background:

• The planar matching problem involves analyzing random matrices with binary entries.
• Understanding the conditions for perfect planar matching structures is crucial in various scientific domains.

Purpose of the Study:

• To determine the critical density of contacts (p(c)) required for perfect planar matching in random matrices.
• To analytically estimate the transition point (p(c)) and its relation to other physical phenomena.

Main Methods:

• Formulation of the planar matching problem using Dyck paths.
• Development of a matrix model for planar contact structures.
• Analytical estimation of the critical density in the thermodynamic limit.

Main Results:

• A perfect planar matching structure exists only above a critical density, p(c).
• An analytical estimation for p(c) was derived, closely matching numerical simulations (p(c)≈0.379).
• The perfect-imperfect matching transition is linked to the molten-glass transition in random RNA secondary structures.

Conclusions:

• The study provides strong evidence that the molten-glass transition at absolute zero (T=0) occurs at the critical density p(c).
• The findings offer insights into the behavior of random RNA secondary structures and phase transitions.