Triviality of the scaling limits of critical Ising and $φ^4$ models with effective dimension at least four
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913812552089600 |
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| author | Panis, Romain |
| author_facet | Panis, Romain |
| contents | We prove that any scaling limit of a critical reflection positive Ising or $φ^4$ model of effective dimension $d_{\text{eff}}$ at least four is Gaussian. This extends the recent breakthrough work of Aizenman and Duminil-Copin -- which demonstrates the corresponding result in the setup of nearest-neighbour interactions in dimension four -- to the case of long-range reflection positive interactions satisfying $d_{\text{eff}}=4$. The proof relies on the random current representation which provides a geometric interpretation of the deviation of the models' correlation functions from Wick's law. When $d=4$, long-range interactions are handled with the derivation of a criterion that relates the speed of decay of the interaction to two different mechanisms that entail Gaussianity: interactions with a sufficiently slow decay induce a faster decay at the level of the model's two-point function, while sufficiently fast decaying interactions force a simpler geometry on the currents which allows to extend nearest-neighbour arguments. When $1\leq d\leq 3$ and $d_{\text{eff}}=4$, the phenomenology is different as long-range effects play a prominent role. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05797 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Triviality of the scaling limits of critical Ising and $φ^4$ models with effective dimension at least four Panis, Romain Probability Mathematical Physics 60G60, 60K35, 82B20, 82B27 We prove that any scaling limit of a critical reflection positive Ising or $φ^4$ model of effective dimension $d_{\text{eff}}$ at least four is Gaussian. This extends the recent breakthrough work of Aizenman and Duminil-Copin -- which demonstrates the corresponding result in the setup of nearest-neighbour interactions in dimension four -- to the case of long-range reflection positive interactions satisfying $d_{\text{eff}}=4$. The proof relies on the random current representation which provides a geometric interpretation of the deviation of the models' correlation functions from Wick's law. When $d=4$, long-range interactions are handled with the derivation of a criterion that relates the speed of decay of the interaction to two different mechanisms that entail Gaussianity: interactions with a sufficiently slow decay induce a faster decay at the level of the model's two-point function, while sufficiently fast decaying interactions force a simpler geometry on the currents which allows to extend nearest-neighbour arguments. When $1\leq d\leq 3$ and $d_{\text{eff}}=4$, the phenomenology is different as long-range effects play a prominent role. |
| title | Triviality of the scaling limits of critical Ising and $φ^4$ models with effective dimension at least four |
| topic | Probability Mathematical Physics 60G60, 60K35, 82B20, 82B27 |
| url | https://arxiv.org/abs/2309.05797 |