On the Phase Diagram of the Random Field Ising Model on the Bethe Lattice

The ferromagnetic Ising model on the Bethe lattice of degree k is considered in the presence of a dichotomous external random field fx= +a and the temperature T>0. We give a description of a part of the phase diagram of this model in the T-a. plane, where we are able to construct limiting Gibbs states and ground states. By comparison with the model with a constant external field we show that for all realizations {={£x= ±a} of the external random field: (i) the Gibbs state is unique for T> Tc ( k > 2 and any a) or for a> 3 (k = 2 and any T); ( i i ) the +-phases coexist in the domain {T< T c , a . < H F ( T ) } , where Tc is the critical temperature and HF(T) is the critical external field in the ferromagnetic Ising model on the Bethe lattice with a constant external field. Then we prove that for almost all £,: ( i i i) the +-phases coexist in a larger domain { T< T c , a < H F ( T ) +e(T)}, where £(T)>0; and (iv) the Gibbs state is unique for 3 > a > 2 at any T. We show that the residual entropy at T—0 is positive for 3 > a > 2, and we give a constructive description of ground states, by so-called dipole configurations.