Spin-models in random fields (RFs) are good representations of many impure materials. Their macroscopic collective behaviour is dominated by the fluctuations in the random fields which accumulate on large scales even if the local field is arbitrarily small. This feature is shared by other weakly disordered models, like flux lines or domain walls in random media. We review some of the main theoretical attempts to describe such systems. A modification of Harris’ argument demonstrates that at the critical point the RF disorder is relevant and that (hyper)scaling must be changed. A domain argument invented by Imry and Ma shows that long-range order is not destroyed by weak RFs in more than d=2 dimensions. This result is supported both by a more refined treatment of the domain argument and by considering the roughness of an isolated domain wall due to the randomness. The wall (or flux line) becomes rough due to disorder but if d>2 the wall remains a well-defined object in RF systems. Different approaches are used to calculate the roughness exponent ζ for walls and lines. Some applications of ζ for the description of type-II superconductors and incommensurate systems are given. More detailed calculations are possible for one-dimensional, Bethe-lattice or the hierarchical Dyson model systems, which confirm as a rule the more approximate treatment of the other sections. In one dimension there is an interesting relation between the statistical mechanics of these models and nonlinear dynamics. Non-classical critical behaviour occurs in RF systems for d<6 and is determined in general by three independent exponents which fulfil certain inequalities. The new exponent θ≡yJ>0 is related to the violation of conventional hyperscaling and is determined by the energy ~H0ξ0 of a correlated region of size ξ. In a renormalization group treatment, the temperature T turns out to be a (dangerous) irrelevant variable which is the most prominent property of the systems considered in this review. The irrelevance of thermal fluctuations on large scales produces metastability and hysteresis effects both in the transition region and in the ordered phase, only briefly considered here. These features occur also in other systems with a disordered T=0 fixed point like in the ordered phase of a spin-glass.
Stochastic Model and Generator for Random Fields with Symmetry Properties: Application to the Mesoscopic Modeling of Elastic Random Media