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