The torus plateau for the high-dimensional Ising model
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arXiv
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| Format: | Preprint |
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2024
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| author | Liu, Yucheng Panis, Romain Slade, Gordon |
| author_facet | Liu, Yucheng Panis, Romain Slade, Gordon |
| contents | We consider the Ising model on a $d$-dimensional discrete torus of volume $r^d$, in dimensions $d>4$ and for large $r$, in the vicinity of the infinite-volume critical point $β_c$. We prove that for $β=β_c- {\rm const}\, r^{-d/2}$ (with a suitable constant) the susceptibility is bounded above and below by multiples of $r^{d/2}$. Additionally, again for $β=β_c- {\rm const}\, r^{-d/2}$, the two-point function has a ``plateau'': it decays like $|x|^{-(d-2)}$ when $|x|$ is small relative to the volume, but for larger $|x|$, it levels off to a constant value of order $r^{-d/2}$. We also prove that at $β=β_c- {\rm const}\, r^{-d/2}$ the renormalised coupling constant is nonzero, which implies a non-Gaussian limit for the average spin. The proof relies on near-critical estimates for the infinite-volume two-point function obtained recently by Duminil-Copin and Panis, and builds upon a strategy proposed by Papathanakos. The random current representation of the Ising model plays a central role in our analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17353 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The torus plateau for the high-dimensional Ising model Liu, Yucheng Panis, Romain Slade, Gordon Mathematical Physics Probability 82B20, 82B27, 60K35 We consider the Ising model on a $d$-dimensional discrete torus of volume $r^d$, in dimensions $d>4$ and for large $r$, in the vicinity of the infinite-volume critical point $β_c$. We prove that for $β=β_c- {\rm const}\, r^{-d/2}$ (with a suitable constant) the susceptibility is bounded above and below by multiples of $r^{d/2}$. Additionally, again for $β=β_c- {\rm const}\, r^{-d/2}$, the two-point function has a ``plateau'': it decays like $|x|^{-(d-2)}$ when $|x|$ is small relative to the volume, but for larger $|x|$, it levels off to a constant value of order $r^{-d/2}$. We also prove that at $β=β_c- {\rm const}\, r^{-d/2}$ the renormalised coupling constant is nonzero, which implies a non-Gaussian limit for the average spin. The proof relies on near-critical estimates for the infinite-volume two-point function obtained recently by Duminil-Copin and Panis, and builds upon a strategy proposed by Papathanakos. The random current representation of the Ising model plays a central role in our analysis. |
| title | The torus plateau for the high-dimensional Ising model |
| topic | Mathematical Physics Probability 82B20, 82B27, 60K35 |
| url | https://arxiv.org/abs/2405.17353 |