Random Magnetic Fields, Supersymmetry, and Negative Dimensions

Abstract

We prove the equivalence, near the critical point, of a $D$-dimensional spin system in a random external magnetic field with a ($D\ensuremath{-}2$)-dimensional spin system in the absence of a magnetic field. This is due to the hidden supersymmetry of the associated stochastic differential equation. We identify a space with one anticommuting coordinate with a space having negative dimensions -2.