RNA-like polymer model: exact calculation on the Bethe lattice

We consider a lattice polymer model (random walk), in which the walk is allowed to visit lattice bonds at most twice. Such a model might have some relevance to describe statistical properties of RNA molecules. In order to mimic base pairing, we assign an attractive energy term to each doubly visited bond, and a further contribution to each pair of consecutive doubly visited bonds. The latter term is expected to mimic the stacking effect, whereas no effect of sequence, that is, of chemical specificity, is taken into account. The phase diagram is worked out exactly on a Bethe lattice, in a grand-canonical formulation. In the single molecule limit, the system undergoes two different phase transitions upon decreasing temperature: a Theta-like collapse from a swollen "coil" state to a "molten" state, with a low fraction of doubly visited bonds, and subsequently to a "paired" state, with empty or doubly visited bonds only. The stacking effect drives the latter transition from second to first order.