RNA matrix models with external interactions and their asymptotic behaviour

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 $n\alpha/L=1$)

💡 Research Summary

The paper extends the well‑known random‑matrix model of RNA secondary structures by introducing an external perturbation that acts on a subset of nucleotides. In the original (unperturbed) model, the partition function’s exponential generating function leads to a universal growth of the number of diagrams (configurations) at fixed chain length L and genus g, scaling as 3^L, while the total number of diagrams summed over all genera grows as exp(√L). The authors modify the model by adding a perturbation term whose strength is α times that of the original interaction and that affects n nucleotides out of the total length L. This results in a multiplicative factor (1 − nα/L) appearing in the exponent of the generating function.

Two limiting cases are examined numerically: (i) the perturbation acts on the whole chain (n = L) and (ii) it acts on a single nucleotide (n = 1). By varying the dimensionless parameter x = nα/L from 0 to 1, the authors track how the genus‑specific diagram counts a_{L,g} and the total diagram count 𝒩 change. The key findings are:

1. Genus‑specific universality – For any fixed L and g, the leading exponential factor evolves from 3^L (x = 0) to (3 − x)^L as x increases, and reaches 2^L when x = 1. This indicates that the external field progressively reduces the combinatorial freedom of base‑pairing, ultimately leaving only two possible configurations per site when the perturbation dominates the whole chain.

2. Total diagram count – The sub‑exponential factor exp(√L) is replaced by exp