  We study pressurised self-avoiding ring polymers in two dimensions using
Monte Carlo simulations, scaling arguments and Flory-type theories, through
models which generalise the model of Leibler, Singh and Fisher [Phys. Rev.
Lett. Vol. 59, 1989 (1987)]. We demonstrate the existence of a thermodynamic
phase transition at a non-zero scaled pressure $\tilde{p}$, where $\tilde{p} =
Np/4\pi$, with the number of monomers $N \rightarrow \infty$ and the pressure
$p \rightarrow 0$, keeping $\tilde{p}$ constant, in a class of such models.
This transition is driven by bond energetics and can be either continuous or
discontinuous. It can be interpreted as a shape transition in which the ring
polymer takes the shape, above the critical pressure, of a regular N-gon whose
sides scale smoothly with pressure, while staying unfaceted below this critical
pressure. In the general case, we argue that the transition is replaced by a
sharp crossover. The area, however, scales with $N^2$ for all positive $p$ in
all such models, consistent with earlier scaling theories.
