  In a recent experiment, Milliken {\em et al.} demonstrated possible evidence
for a Luttinger liquid through measurements of the tunneling conductance
between edge states in the $\nu=1/3$ quantum Hall plateau. However, at low
temperatures, a discrepancy exists between the theoretical predictions based on
Luttinger liquid theory and experiment. We consider the possibility that this
is due to long-range Coulomb interactions which become dominant at low
temperatures. Using renormalization group methods, we calculate the cross-over
behaviour from Luttinger liquid to the Coulomb interaction dominated regime.
The cross-over behaviour thus obtained seems to resolve one of the
discrepancies, yielding good agreement with experiment.
