Phase transitions in low-dimensional long-range random field Ising models
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909460197277696 |
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| author | Ding, Jian Huang, Fenglin Maia, João |
| author_facet | Ding, Jian Huang, Fenglin Maia, João |
| contents | We consider the long-range random field Ising model in dimension $d = 1, 2$, whereas the long-range interaction is of the form $J_{xy} = |x-y|^{-α}$ with $1< α< 3/2$ for $d=1$ and with $2 < α\leq 3$ for $d = 2$. Our main results establish phase transitions in these regimes. In one dimension, we employ a Peierls argument with some novel modification, suitable for dealing with the randomness coming from the external field; in two dimensions, our proof follows that of Affonso, Bissacot, and Maia (2023) with some adaptations, but new ideas are required in the critical case of $α=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19281 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Phase transitions in low-dimensional long-range random field Ising models Ding, Jian Huang, Fenglin Maia, João Probability Mathematical Physics 82B44, 60K35, 82B26 We consider the long-range random field Ising model in dimension $d = 1, 2$, whereas the long-range interaction is of the form $J_{xy} = |x-y|^{-α}$ with $1< α< 3/2$ for $d=1$ and with $2 < α\leq 3$ for $d = 2$. Our main results establish phase transitions in these regimes. In one dimension, we employ a Peierls argument with some novel modification, suitable for dealing with the randomness coming from the external field; in two dimensions, our proof follows that of Affonso, Bissacot, and Maia (2023) with some adaptations, but new ideas are required in the critical case of $α=3$. |
| title | Phase transitions in low-dimensional long-range random field Ising models |
| topic | Probability Mathematical Physics 82B44, 60K35, 82B26 |
| url | https://arxiv.org/abs/2412.19281 |