A simple probabilistic method is used to solve exactly the problem of hysteresis and avalanches in the random field Ising model on a Bethe lattice in the limit of zero temperature and zero frequency of the driving field. For a Gaussian distribution of the random field, we find a discontinuity in the response of the system to the applied field, and non-equilibrium critical phenomena on lattices of coordination number z > 3. This result is discussed in the context of numerical simulations of the model on random graphs and d-dimensional periodic lattices.

North Eastern Hill University, Shillong, India