Metastable states and T=0 hysteresis in the random-field Ising 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 an external magnetic field H. We compute the number of single-spin-flip stable configurations with a given magnetization m and study the connection between the distribution of these metastable states in the H - m plane ( focusing on the region where the number is exponentially large) and the shape of the saturation hysteresis loop obtained by cycling the field between -infinity and +infinity at T = 0. The annealed complexity Sigma(A)(m, H) is calculated for z = 2, 3, 4 and the quenched complexity Sigma(Q)(m, H) for z = 2. We prove explicitly for z = 2 that the contour Sigma(Q)(m, H) = 0 coincides with the saturation loop. On the other hand, we show that Sigma(A)(m, H) is irrelevant for describing, even qualitatively, the observable hysteresis properties of the system.