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

Abstract

The ferromagnetic Ising model on the Bethe lattice of degree k is considered in the presence of a dichotomous external random field ξ _x = ±α and the temperature T≥0. We give a description of a part of the phase diagram of this model in the T−α 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 ξ = {ξ _x = ±α} of the external random field: (i) the Gibbs state is unique for T > T _c (k ≥ 2 and any α) or for α > 3 (k = 2 and any T); (ii) the ±-phases coexist in the domain {T < T _c, α ≤ 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 field. Then we prove that for almost all ξ: (iii) the ±-phases coexist in a larger domain {T < T _c, α ≤H ^F(T) + ε(T)}, where ε(T)>0; and (iv) the Gibbs state is unique for 3≥α≥2 at any T. We show that the residual entropy at T = 0 is positive for 3≥α≥2, and we give a constructive description of ground states, by so-called dipole configurations.