Using Monte Carlo techniques, we show that vortex strings are responsible for the phase transition in the three-dimensional planar model. The phase structure for a generalized planar model with vortex suppression is determined. This latter model has the unusual feature of long-range order in a ground state with finite, disordering entropy.

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Using Monte Carlo techniques, we study the magnetic susceptibility and specific heat of the two-dimensional step model. Contrary to some theoretical expectations and numerical results, the finite-size-scale analysis of the susceptibility sho~s evidence for a possible low-temperature phase with in6nite correlation lengths and with a nondivergent speci6c heat. The absence in this model of long-lived metastable states is also analyzed.