Scaling, phase transition and genus distribution functions in matrix models of RNA with linear external interactions

A linear external perturbation is introduced in the action of the partition function of the random matrix model of RNA [ It is seen that (i). the perturbation distinguishes between paired and unpaired bases in that there are structural changes, from unpaired and paired base structures (0 ≤ α < 1) to completely paired base structures (α = 1), as the perturbation parameter α approaches 1 (α is the ratio of interaction strengths of original and perturbed terms in the action of the partition function), (ii). the genus distributions exhibit small differences for small even and odd lengths L, (iii). the partition function of the linear interacting matrix model is related via a scaling formula to the re-scaled partition function of the random matrix model of RNA, (iv). the free energy and specific heat are plotted as functions of L, α and temperature T and their first derivative with respect to α is plotted as a function of α. The free energy shows a phase transition at α = 1 for odd (both small and large) lengths and for even lengths the transition at α = 1 gets sharper and sharper as more pseudoknots are included (that is for large lengths).