  We analyze the demagnetization properties of the random-field Ising model on
the Bethe lattice focusing on the beahvior near the disorder induced phase
transition. We derive an exact recursion relation for the magnetization and
integrate it numerically. Our analysis shows that demagnetization is possible
only in the continous high disorder phase, where at low field the loops are
described by the Rayleigh law. In the low disorder phase, the saturation loop
displays a discontinuity which is reflected by a non vanishing magnetization
m_\infty after a series of nested loops. In this case, at low fields the loops
are not symmetric and the Rayleigh law does not hold.
