Non-constant Ground Configurations in the Disordered Ferromagnet

Abstract

The disordered ferromagnet is a disordered version of the ferromagnetic Ising model in which the coupling constants are non-negative quenched random. A ground configuration is an infinite-volume configuration whose energy cannot be reduced by finite modifications. It is a long-standing challenge to ascertain whether the disordered ferromagnet on the \(\mathbb {Z}^D\) lattice admits non-constant ground configurations. We answer this affirmatively in dimensions \(D\ge 4\), when the coupling constants are sampled independently from a sufficiently concentrated distribution. The obtained ground configurations are further shown to be translation-covariant with respect to \(\mathbb {Z}^{D-1}\) translations of the disorder. Our result is proved by showing that the finite-volume interface formed by Dobrushin boundary conditions is localized, and converges to an infinite-volume interface. This may be expressed in purely combinatorial terms, as a result on the fluctuations of certain minimal cutsets in the lattice \(\mathbb {Z}^D\) endowed with independent edge capacities.