Non-constant ground configurations in the disordered ferromagnet
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916988565061632 |
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| author | Bassan, Michal Gilboa, Shoni Peled, Ron |
| author_facet | Bassan, Michal Gilboa, Shoni Peled, Ron |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06437 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-constant ground configurations in the disordered ferromagnet Bassan, Michal Gilboa, Shoni Peled, Ron Mathematical Physics Probability 82B44, 60K37, 60K35, 82B41 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. |
| title | Non-constant ground configurations in the disordered ferromagnet |
| topic | Mathematical Physics Probability 82B44, 60K37, 60K35, 82B41 |
| url | https://arxiv.org/abs/2309.06437 |