Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910565169889280 |
|---|---|
| author | Affonso, Lucas Bissacot, Rodrigo Endo, Eric O. Handa, Satoshi |
| author_facet | Affonso, Lucas Bissacot, Rodrigo Endo, Eric O. Handa, Satoshi |
| contents | Inspired by Fröhlich-Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on $\mathbb{Z}^d$, $d\geq 2$. The argument, which is based on a multi-scale analysis, works for the sharp region $α>d$ and improves previous results obtained by Park for $α>3d+1$, and by Ginibre, Grossmann, and Ruelle for $α> d+1$, where $α$ is the power of the coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomially decaying magnetic field with power $δ>0$ as $h^*|x|^{-δ}$, where $h^* >0$. For $d<α<d+1$, the phase transition occurs when $δ>α-d$, and when $h^*$ is small enough over the critical line $δ=α-d$. For $α\geq d+1$, $δ>1$ is enough to prove the phase transition, and for $δ=1$ we have to ask $h^*$ small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_06103 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields Affonso, Lucas Bissacot, Rodrigo Endo, Eric O. Handa, Satoshi Mathematical Physics Statistical Mechanics Probability 82B05, 82B26, 82B20, 82Bxx Inspired by Fröhlich-Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on $\mathbb{Z}^d$, $d\geq 2$. The argument, which is based on a multi-scale analysis, works for the sharp region $α>d$ and improves previous results obtained by Park for $α>3d+1$, and by Ginibre, Grossmann, and Ruelle for $α> d+1$, where $α$ is the power of the coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomially decaying magnetic field with power $δ>0$ as $h^*|x|^{-δ}$, where $h^* >0$. For $d<α<d+1$, the phase transition occurs when $δ>α-d$, and when $h^*$ is small enough over the critical line $δ=α-d$. For $α\geq d+1$, $δ>1$ is enough to prove the phase transition, and for $δ=1$ we have to ask $h^*$ small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field. |
| title | Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields |
| topic | Mathematical Physics Statistical Mechanics Probability 82B05, 82B26, 82B20, 82Bxx |
| url | https://arxiv.org/abs/2105.06103 |