Lower Critical Dimension of the Random-Field Ising Model

Abstract

A new argument is given for a lower critical dimension ${d}_{l}=2$ for the Ising model in a random magnetic field. It forms the basis for a proof that the three-dimensional model exhibits long-range order at zero temperature and small disorder. This settles the controversy between the values ${d}_{l}=2$ and ${d}_{l}=3$.