We study a matrix model of RNA in which an external perturbation on $n$ nucleotides is introduced in the action of the partition function of the polymer chain. The effect of the perturbation appears in the exponential generating function of the partition function as a factor $\text{exp}\left(1-n\alpha /L\right)$ (where $\alpha$ is the ratio of strengths of the original to the perturbed term and $L$ is the length of the chain). The asymptotic behavior of the genus distribution functions as a function of length for the matrix model with interaction is analyzed numerically for all $n\le L$. It is found that as $n\alpha /L$ is increased from 0 to 1, the term $3^L$ in the number of diagrams $a_{L,g,\alpha }^{\prime }$ at a fixed length $L$, genus $g$ and $\alpha$, goes to $2^L$ [${\left(3-\frac{n\alpha }{L}\right)}^L$ for any $n\alpha /L$] and the total number of diagrams ${\mathcal{N}}_{\alpha }^{\prime }$ at a fixed length $L$ and $\alpha$ but independent of genus $g$, undergoes a change in the factor $\text{exp}\left(\sqrt{L}\right)$ to 1 ($\text{exp}\left[\right(1-n\alpha /L\left)\sqrt{L}\right]$ for any $n\alpha /L$). However the exponent $L$ of the dominant length dependent term in $a_{L,g,\alpha }^{\prime }$ stays unchanged. Hence the universality is robust to changes in the interaction $\left(\alpha \right)$. The distribution functions also exhibit unusual behavior at small lengths.

• Received 20 August 2008

DOI:https://doi.org/10.1103/PhysRevE.79.061903