A-localized states for clock models on trees and their extremal decomposition into glassy states
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917839073443840 |
|---|---|
| author | Kuelske, Christof Schubert, Niklas |
| author_facet | Kuelske, Christof Schubert, Niklas |
| contents | We consider $\mathbb{Z}_q$-valued clock models on a regular tree, for general classes of ferromagnetic nearest neighbor interactions which have a discrete rotational symmetry. It has been proved recently that, at strong enough coupling, families of homogeneous Markov chain Gibbs states $μ_A$ coexist whose single-site marginals concentrate on $A\subset \mathbb{Z}_q$, and which are not convex combinations of each other [AbHeKuMa24]. In this note, we aim at a description of the extremal decomposition of $μ_A$ for $|A|\geq 2$ into all extremal Gibbs measures, which may be spatially inhomogeneous. First, we show that in regimes of very strong coupling, $μ_A$ is not extremal. Moreover, $μ_A$ possesses a single-site reconstruction property which holds for spin values sent from the origin to infinity, when these initial values are chosen from $A$. As our main result, we show that $μ_A$ decomposes into uncountably many extremal inhomogeneous states. The proof is based on multi-site reconstruction which allows to derive concentration properties of branch overlaps. Our method is based on a new good site/bad site decomposition adapted to the $A$-localization property, together with a coarse graining argument in local state space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10271 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A-localized states for clock models on trees and their extremal decomposition into glassy states Kuelske, Christof Schubert, Niklas Probability 60K35, 82B20, 82B44 G.3 We consider $\mathbb{Z}_q$-valued clock models on a regular tree, for general classes of ferromagnetic nearest neighbor interactions which have a discrete rotational symmetry. It has been proved recently that, at strong enough coupling, families of homogeneous Markov chain Gibbs states $μ_A$ coexist whose single-site marginals concentrate on $A\subset \mathbb{Z}_q$, and which are not convex combinations of each other [AbHeKuMa24]. In this note, we aim at a description of the extremal decomposition of $μ_A$ for $|A|\geq 2$ into all extremal Gibbs measures, which may be spatially inhomogeneous. First, we show that in regimes of very strong coupling, $μ_A$ is not extremal. Moreover, $μ_A$ possesses a single-site reconstruction property which holds for spin values sent from the origin to infinity, when these initial values are chosen from $A$. As our main result, we show that $μ_A$ decomposes into uncountably many extremal inhomogeneous states. The proof is based on multi-site reconstruction which allows to derive concentration properties of branch overlaps. Our method is based on a new good site/bad site decomposition adapted to the $A$-localization property, together with a coarse graining argument in local state space. |
| title | A-localized states for clock models on trees and their extremal decomposition into glassy states |
| topic | Probability 60K35, 82B20, 82B44 G.3 |
| url | https://arxiv.org/abs/2411.10271 |