We show that tunnelling in the quantum Hall regime acquires the form o
f thermal activation above a crossover temperature T-1. The width of t
he conductance peak sigma(xx) between Hall plateaus scales then with t
emperature as T-kappa with kappa = 3/7 at T < T-1 and crosses over to
kappa = 6/7 at T > T-1. We find that T-1 decreases for longer-range po
tentials or for higher magnetic fields, hence at a given temperature r
ange the apparent kappa should decrease for samples with shorter-range
scattering or in a given sample at higher Landau levels. These result
s are consistent with data showing 0.3 less than or similar to kappa l
ess than or similar to 0.8 for different Landau levels and for the rem
arkable difference between InxGa1-xAs/InP and GaAs/AlxGa1-xAs heterost
ructures.