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.