  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 dilute solution (single molecule) limit,
the system undergoes two different phase transitions, upon decreasing
temperature: a Thetalike 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.
