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 xi(x)= +/-alpha and the temperature T greater than or equal to 0. We give a description of a part of the phase diagram of this model in the T-alpha 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 ail realizations xi= {xi(x) = +/-alpha} of the external random field: (i) the Gibbs stale is unique for T > T-c(k greater than or equal to 2 and any alpha) or for alpha > 3 (k = 2 and any T); (ii) the +/--phases coexist in the domain {T< T-c, alpha less than or equal to H-F(T)}, where T-c is the critical temperature and H-F(T) is the critical external field in the ferromagnetic Ising model on the Bethe lattice with a constant external held. Then we prove that for almost all xi: (iii) the +/--phases coexist in a larger domain {T< T-c, alpha less than or equal to H-F(T) + epsilon(T)}, where epsilon(T) > 0; and (iv) the Gibbs state is unique for 3 greater than or equal to alpha greater than or equal to 2 at any T. We show that the residual entropy at T=0 is positive for 3 greater than or equal to alpha greater than or equal to 2, and we give a constructive description of ground states, by so-called dipole configurations.