Binary fluid in a random medium and dimensional transmutation

John Cardy

Published 2023-05-23Version 1

We propose a solution to the puzzle of dimensional reduction in the random field Ising model by studying the equivalent binary fluid in a random medium, and inverting the question by asking: to what random problem in $D\!=\!d+2$ dimensions does a pure system in $d$ dimensions correspond? For a model with purely repulsive interactions of finite range, 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 $\propto\lambda$, in the limit as $\lambda\to\infty$. There is no conflict with rigorous results that the finite range model with locally correlated disorder orders in $D=3$. Our argument avoids the use of replicas and perturbative field theory, instead being based on convergent cluster expansions. Although the result may be seen as a consequence of Parisi-Sourlas supersymmetry, it follows more directly from Kirchhoff's matrix-tree theorem.