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 $-\infty$ and $+\infty$ 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.