  We consider the single-spin-flip dynamics of the random-field Ising model on
a Bethe lattice at zero temperature in the presence of a uniform external
field. We determine the average magnetization as the external field is varied
from minus infinity to plus infinity by setting up the self-consistent field
equations, which we show are exact in this case. We find that for a
3-coordinated Bethe lattice, there is no jump discontinuity in magnetization
for arbitrarily small gaussian disorder, but the discontinuity is present for
larger coordination numbers. We have checked our results by Monte Carlo
simulations employing a technique for simulating classical interacting systems
on the Bethe lattice which avoids surface effects altogether.
