It is also interesting to study the behavior directly in two dimensions, in which . This is the lower critical dimension, and may therefore be purely thermal or contain a ``marginal'' glassy phase.
While the general belief is that the latter holds (in this phase-slip free model), there is as yet no agreement in detail on the properties of this phase. A RG treatment was performed some time ago by Cardy and Ostlund in a different context, and appears to provide a sound description of a finite temperature fixed line describing a glassy phase. This predicts fluctuations of , and more recently has been extended to probe the susceptibility and sensitivity to temperature changes. Various RSB approaches, however, are in conflict with these results, and generally give only variations of the phase. Numerical simulations performed so far have not observed the predicted fluctuations, and in fact disagree in other ways with the expected signatures of the glass transition. The resolution of these discrepancies is unclear.