RNA matrix models with external interactions and their asymptotic behavior.

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 exp(1-nalpha/L) (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<or=L. It is found that as nalpha/L is increased from 0 to 1, the term 3L in the number of diagrams a'L,g,alpha at a fixed length L, genus g and alpha, goes to 2L [(3-nalpha/L)L for any nalpha/L] and the total number of diagrams Nalpha' at a fixed length L and alpha but independent of genus g, undergoes a change in the factor exp(sqrt[L]) to 1 (exp[(1-nalpha/L)sqrt[L]] for any nalpha/L). However the exponent L of the dominant length dependent term in a'L,g,alpha stays unchanged. Hence the universality is robust to changes in the interaction (alpha). The distribution functions also exhibit unusual behavior at small lengths.