The supercritical phase of the $φ^4$ model is well behaved
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915312826318848 |
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| author | Gunaratnam, Trishen S. Panagiotis, Christoforos Panis, Romain Severo, Franco |
| author_facet | Gunaratnam, Trishen S. Panagiotis, Christoforos Panis, Romain Severo, Franco |
| contents | In this article, we analyse the $φ^4$ model on $\mathbb Z^d$ in the supercritical regime $β> β_c$. We consider a random cluster representation of the $φ^4$ model, which corresponds to an Ising random cluster model on a random environment. We prove that the supercritical phase of this percolation model on $\mathbb Z^d$ ($d\geq 2$) is well behaved in the sense that, for every $β>β_c$, local uniqueness of macroscopic clusters occurs with high probability, uniformly in the boundary conditions. This result provides the basis for renormalisation techniques used to study several fine properties of the supercritical phase. As applications, we prove surface order exponential bounds for the (lower) large deviations of the empirical magnetisation as well as for the spectral gaps of dynamical $φ^4$ models in the entire supercritical regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The supercritical phase of the $φ^4$ model is well behaved Gunaratnam, Trishen S. Panagiotis, Christoforos Panis, Romain Severo, Franco Probability Mathematical Physics 82B20, 60K35 In this article, we analyse the $φ^4$ model on $\mathbb Z^d$ in the supercritical regime $β> β_c$. We consider a random cluster representation of the $φ^4$ model, which corresponds to an Ising random cluster model on a random environment. We prove that the supercritical phase of this percolation model on $\mathbb Z^d$ ($d\geq 2$) is well behaved in the sense that, for every $β>β_c$, local uniqueness of macroscopic clusters occurs with high probability, uniformly in the boundary conditions. This result provides the basis for renormalisation techniques used to study several fine properties of the supercritical phase. As applications, we prove surface order exponential bounds for the (lower) large deviations of the empirical magnetisation as well as for the spectral gaps of dynamical $φ^4$ models in the entire supercritical regime. |
| title | The supercritical phase of the $φ^4$ model is well behaved |
| topic | Probability Mathematical Physics 82B20, 60K35 |
| url | https://arxiv.org/abs/2501.05353 |