Metastable states and T=0 hysteresis in the random-fieldIsing model on random graphs

We study the ferromagnetic random-field Ising model on random graphs of fixed connectivity z (Bethe lattice) in the presence of anexternal magnetic field H. We compute the number of single-spin-flipstable configurations with a given magnetization m and study theconnection between the distribution of these metastable states in the H-m plane(focusing on the region where the number is exponentially large) andthe shape of the saturation hysteresis loop obtained by cycling thefield between - and + at T=0. The annealed complexity A(m,H) iscalculated for z=2,3,4 and the quenched complexity Q(m,H) forz=2. We prove explicitly for z=2 that the contourQ(m,H)=0 coincides with the saturation loop. On the other hand, we show that A(m,H) is irrelevant for describing, even qualitatively, the observable hysteresis properties of the system.