RNA matrix models with external interactions and their asymptotic behaviour

TL;DR

This paper investigates the asymptotic behavior of RNA matrix models with external perturbations, revealing how the number of diagrams and their universality change with perturbation strength, especially at extreme cases where the perturbation affects all or one nucleotide.

Contribution

It introduces a modified RNA matrix model incorporating external interactions and analyzes its asymptotic properties numerically for different perturbation scenarios.

Findings

Number of diagrams transitions from 3^L to (3 - nα/L)^L as perturbation increases.

Total number of diagrams' asymptotic growth changes from exp^{√L} to exp^{(1 - nα/L)√L}.

Universality class shifts depending on the perturbation strength and affected nucleotides.

Abstract

We study a matrix model of RNA in which an external perturbation acts on n nucleotides of the polymer chain. The effect of the perturbation appears in the exponential generating function of the partition function as a factor $(1-\frac{n\alpha}{L})$ [where $\alpha$ is the ratio of strengths of the original to the perturbed term and L is length of the chain]. The asymptotic behaviour of the genus distribution functions for the extended matrix model are analyzed numerically when (i) $n=L$ and (ii) $n=1$. In these matrix models of RNA, as $n\alpha/L$ is increased from 0 to 1, it is found that the universality of the number of diagrams $a_{L, g}$ at a fixed length L and genus g changes from $3^{L}$ to $(3-\frac{n\alpha}{L})^{L}$ ($2^{L}$ when $n\alpha/L=1$) and the asymptotic expression of the total number of diagrams $\cal N$ at a fixed length L but independent of genus g, changes in the factor $\exp^{\sqrt{L}}$ to $\exp^{(1-\frac{n\alpha}{L})\sqrt{L}}$ ($exp^{0}=1$ when $n\alpha/L=1$)