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Non-constant ground configurations in the disordered ferromagnet

Michal Bassan, Shoni Gilboa, Ron Peled

TL;DR

The paper proves that in dimensions $D\ge 4$ the disordered ferromagnet with suitably concentrated disorder admits non-constant ground configurations and that these configurations can be chosen to be covariant under translations in the first $D-1$ coordinates. The authors combine Dobrushin boundary conditions with a novel shift-based localization approach, coupling-Dobrushin energy gaps, and a multi-scale chaining argument to demonstrate that the finite-volume Dobrushin interface localizes near the reference plane and converges to an infinite-volume interface. Central to the method are the notions of disorder shifts $\tau$, a projected set $\tilde A$ capturing interface features, and admissible shifts whose energetic impact is controlled via coarse/fine grainings and layering bounds; concentration inequalities for energy differences play a key role. The results extend to an anisotropic disordered ferromagnet and yield non-constant covariant ground configurations under reduced symmetry groups, with quantitative rates of convergence and decay of correlations in the infinite-volume limit. Overall, the work rigorously confirms long-standing physics predictions about interface localization and ground-state non-uniqueness in high dimensions for concentrated disorder, and it provides a robust combinatorial framework via minimal cutsets and grainings relevant to related percolation and SOS-type models.

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.

Non-constant ground configurations in the disordered ferromagnet

TL;DR

The paper proves that in dimensions the disordered ferromagnet with suitably concentrated disorder admits non-constant ground configurations and that these configurations can be chosen to be covariant under translations in the first coordinates. The authors combine Dobrushin boundary conditions with a novel shift-based localization approach, coupling-Dobrushin energy gaps, and a multi-scale chaining argument to demonstrate that the finite-volume Dobrushin interface localizes near the reference plane and converges to an infinite-volume interface. Central to the method are the notions of disorder shifts , a projected set capturing interface features, and admissible shifts whose energetic impact is controlled via coarse/fine grainings and layering bounds; concentration inequalities for energy differences play a key role. The results extend to an anisotropic disordered ferromagnet and yield non-constant covariant ground configurations under reduced symmetry groups, with quantitative rates of convergence and decay of correlations in the infinite-volume limit. Overall, the work rigorously confirms long-standing physics predictions about interface localization and ground-state non-uniqueness in high dimensions for concentrated disorder, and it provides a robust combinatorial framework via minimal cutsets and grainings relevant to related percolation and SOS-type models.

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 lattice admits non-constant ground configurations. We answer this affirmatively in dimensions , 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 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 endowed with independent edge capacities.
Paper Structure (63 sections, 66 theorems, 382 equations, 1 figure)

This paper contains 63 sections, 66 theorems, 382 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Localization predictions
  4. Disordered Solid-On-Solid model
  5. Long-range order in the random-field Ising model
  6. Law of large numbers and large deviations of the ground energy
  7. Ground energy fluctuations
  8. Concentrated disorder distributions
  9. Number of ground configurations
  10. Translation-covariant ground metastates
  11. The Dobrushin interface in other settings
  12. Overview of the proof
  13. The random-field Ising model
  14. The disordered Solid-On-Solid model
  15. The disordered ferromagnet
  16. ...and 48 more sections

Key Result

Theorem 1.2

There exists $c>0$ such that the following holds in dimensions $D\ge 4$. Consider the (isotropic) disordered ferromagnet with disorder distribution $\nu$. If $\min(\mathop{\mathrm{supp}}\nolimits(\nu))>0$, $\nu$ has no atoms and then the disordered ferromagnet admits non-constant ground configurations.

Figures (1)

  • Figure 1: An illustration of the corresponding $V_{\sigma_{0}}$ and $\tilde{A}$, given a fixed function $I_{\sigma_0}$, vertex $v$ and set $E$ (of two vertices). Notice that the innermost cross-hatched component is $A_{v}$.

Theorems & Definitions (126)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.7: Localization of Dobrushin interface
  • Remark 1.8: Combinatorial interpretation
  • Theorem 1.9: Convergence
  • Corollary 1.10
  • Theorem 1.11
  • Remark 1.12
  • ...and 116 more