Zero-temperature Hysteresis in Random-field Ising Model on a Bethe   Lattice
  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. 
Deepak Dhar, Prabodh Shukla and James P. Sethna