  We propose a solution to the puzzle of dimensional reduction in the random
field Ising model, inverting the question and asking: to what random problem in
$D=d+2$ dimensions does a pure system in $d$ dimensions correspond? We consider
two models: a continuum binary fluid, and a lattice gas which maps exactly onto
an Ising model. In both cases we show that the mean density and other
observables are equal to those of a similar model in $D$ dimensions, but with
interactions and correlated disorder in the extra two dimensions of range
$\propto l$, in the limit as $l\to\infty$. There is no conflict with rigorous
results that the finite range model with locally correlated disorder orders in
$D=3$. Our arguments avoid the use of replicas and perturbative field theory,
instead being based on convergent cluster expansions, which, for the lattice
gas, may be extended all the way to the critical point by virtue of the
Lee-Yang theorem. Although the results may be viewed as a consequence of
Parisi-Sourlas supersymmetry, they follow more directly from Kirchhoff's
matrix-tree theorem.
