Dimensional reduction with correlated random fields. A superspace renormalization-group calculation

We consider phase transitions in the presence of random magnetic fields, with long-range correlations such that 〈h(0)h(r)〉~1r^d-σ. The upper critical dimension is 6+σ. A superspace renormalization-group calculation is carried out to order ε², where ε=6+σ-d. To order ε, the critical properties are that of a pure system in d-2-σ dimensions. This dimensional reduction is also justified by physical arguments concerning the lower critical dimensionality, and the hyperscaling relationship. However, it fails at order ε².