Elusive Spin-Glass Phase in the Random Field Ising Model

We show rigorously that the spin-glass susceptibility in the random field Ising model is always bounded by the ferromagnetic susceptibility, and therefore that no spin-glass phase can be present at equilibrium out of the ferromagnetic critical line. When the magnetization is, however, fixed to values smaller than the equilibrium value, a spin-glass phase can exist, as we show explicitly on the Bethe lattice.