Non-constant ground configurations in the disordered ferromagnet

TL;DR

This paper proves the existence of non-constant ground configurations in high-dimensional disordered ferromagnets with random couplings, using interface localization and combinatorial methods.

Contribution

It establishes the existence of non-constant ground states in disordered ferromagnets for dimensions D≥4 with certain coupling distributions, a long-standing open problem.

Findings

Non-constant ground configurations exist in D≥4.

Ground configurations are translation-covariant under Z^{D-1}.

Finite-volume interfaces converge to infinite-volume interfaces.

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.