Infinite-volume states with irreducible localization sets for gradient models on trees
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911965258973184 |
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| author | Abbondandolo, Alberto Henning, Florian Kuelske, Christof Majer, Pietro |
| author_facet | Abbondandolo, Alberto Henning, Florian Kuelske, Christof Majer, Pietro |
| contents | We consider general classes of gradient models on regular trees with values in a countable Abelian group $S$ such as $\mathbb{Z}$ or $\mathbb{Z}_q$, in regimes of strong coupling (or low temperature). This includes unbounded spin models like the p-SOS model and finite-alphabet clock models. We prove the existence of families of distinct homogeneous tree-indexed Markov chain Gibbs states $μ_A$ whose single-site marginals concentrate on a given finite subset $A \subset S$ of spin values, under a strong coupling condition for the interaction, depending only on the cardinality $\vert A \vert$ of $A$. The existence of such states is a new and robust phenomenon which is of particular relevance for infinite spin models. These states are not convex combinations of each other, and in particular the states with $\vert A \vert \geq 2$ can not be decomposed into homogeneous Markov-chain Gibbs states with a single-valued concentration center. As a further application of the method we obtain moreover the existence of new types of $\mathbb{Z}$-valued gradient Gibbs states, whose single-site marginals do not localize, but whose correlation structure depends on the finite set $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_05398 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Infinite-volume states with irreducible localization sets for gradient models on trees Abbondandolo, Alberto Henning, Florian Kuelske, Christof Majer, Pietro Probability 82B26 (primary), 60K35 (secondary) We consider general classes of gradient models on regular trees with values in a countable Abelian group $S$ such as $\mathbb{Z}$ or $\mathbb{Z}_q$, in regimes of strong coupling (or low temperature). This includes unbounded spin models like the p-SOS model and finite-alphabet clock models. We prove the existence of families of distinct homogeneous tree-indexed Markov chain Gibbs states $μ_A$ whose single-site marginals concentrate on a given finite subset $A \subset S$ of spin values, under a strong coupling condition for the interaction, depending only on the cardinality $\vert A \vert$ of $A$. The existence of such states is a new and robust phenomenon which is of particular relevance for infinite spin models. These states are not convex combinations of each other, and in particular the states with $\vert A \vert \geq 2$ can not be decomposed into homogeneous Markov-chain Gibbs states with a single-valued concentration center. As a further application of the method we obtain moreover the existence of new types of $\mathbb{Z}$-valued gradient Gibbs states, whose single-site marginals do not localize, but whose correlation structure depends on the finite set $A$. |
| title | Infinite-volume states with irreducible localization sets for gradient models on trees |
| topic | Probability 82B26 (primary), 60K35 (secondary) |
| url | https://arxiv.org/abs/2302.05398 |