Topological transition in disordered planar matching: combinatorial arcs expansion

TL;DR

This paper analyzes a disordered planar matching model, revealing a topological phase transition at a critical density, and introduces a combinatorial arcs expansion method for accurate analytical estimation, with applications to RNA structures.

Contribution

It develops a combinatorial arcs expansion technique to analytically estimate the critical point in a disordered planar matching model, linking it to RNA secondary structure modeling.

Findings

Identified a topological phase transition at a critical density threshold.

Developed an arcs expansion method for analytical estimation.

Applied the model to RNA secondary structure representations.

Abstract

In this paper, we investigate analytically the properties of the disordered Bernoulli model of planar matching. This model is characterized by a topological phase transition, yielding complete planar matching solutions only above a critical density threshold. We develop a combinatorial procedure of arcs expansion that explicitly takes into account the contribution of short arcs, and allows to obtain an accurate analytical estimation of the critical value by reducing the global constrained problem to a set of local ones. As an application to a toy representation of the RNA secondary structures, we suggest generalized models that incorporate a one-to-one correspondence between the contact matrix and the RNA-type sequence, thus giving sense to the notion of effective non-integer alphabets.