Rayleigh loops in the random-field ising model on the bethe lattice

We analyze the demagnetization properties of the random-field Ising model on the Bethe lattice focusing on the behavior 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 continuous 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 that is reflected by a nonvanishing magnetization m(infinity) after a series of nested loops. In this case, at low fields the loops are not symmetric and the Rayleigh law does not hold.