The T=0 random-field Ising model on a Bethe lattice with large   coordination number: hysteresis and metastable states
  In order to elucidate the relationship between rate-independent hysteresis and metastability in disordered systems driven by an external field, we study the Gaussian RFIM at T=0 on regular random graphs (Bethe lattice) of finite connectivity z and compute to O(1/z) (i.e. beyond mean-field) the quenched complexity associated with the one-spin-flip stable states with magnetization m as a function of the magnetic field H. When the saturation hysteresis loop is smooth in the thermodynamic limit, we find that it coincides with the envelope of the typical metastable states (the quenched complexity vanishes exactly along the loop and is positive everywhere inside). On the other hand, the occurence of a jump discontinuity in the loop (associated with an infinite avalanche) can be traced back to the existence of a gap in the magnetization of the metastable states for a range of applied field, and the envelope of the typical metastable states is then reentrant. These findings confirm and complete earlier analytical and numerical studies. 
M.L. Rosinberg, G. Tarjus, and F.J. Perez-Reche