Abstract
We show that the three-dimensional Ising model coupled to a small random magnetic field is ordered at low temperatures. This means that the lower critical dimension,dl for the theory isdl≦2, settling a long controversy on the subject. Our proof is based on an exact Renormalization Group (RG) analysis of the system. This analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of Imry and Ma. The RG acts in the space of Ising models and in the space of random field distributions, driving the former to zero temperature and the latter to zero variance.
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