Random tangled currents for $φ^4$: translation invariant Gibbs measures and continuity of the phase transition
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2022
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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 | We prove that the set of automorphism invariant Gibbs measures for the $φ^4$ model on graphs of polynomial growth has at most two extremal measures at all values of $β$. We also give a sufficient condition to ensure that the set of all Gibbs measures is a singleton. As an application, we show that the spontaneous magnetisation of the nearest-neighbour $φ^4$ model on $\mathbb{Z}^d$ vanishes at criticality for $d\geq 3$. The analogous results were established for the Ising model in the seminal works of Aizenman, Duminil-Copin, and Sidoravicius (Comm. Math. Phys., 2015), and Raoufi (Ann. Prob., 2020) using the so-called random current representation introduced by Aizenman (Comm. Math. Phys., 1982). One of the main contributions of this paper is the development of a corresponding geometric representation for the $φ^4$ model called the random tangled current representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_00319 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Random tangled currents for $φ^4$: translation invariant Gibbs measures and continuity of the phase transition Gunaratnam, Trishen S. Panagiotis, Christoforos Panis, Romain Severo, Franco Probability Mathematical Physics 60K35, 82B20 We prove that the set of automorphism invariant Gibbs measures for the $φ^4$ model on graphs of polynomial growth has at most two extremal measures at all values of $β$. We also give a sufficient condition to ensure that the set of all Gibbs measures is a singleton. As an application, we show that the spontaneous magnetisation of the nearest-neighbour $φ^4$ model on $\mathbb{Z}^d$ vanishes at criticality for $d\geq 3$. The analogous results were established for the Ising model in the seminal works of Aizenman, Duminil-Copin, and Sidoravicius (Comm. Math. Phys., 2015), and Raoufi (Ann. Prob., 2020) using the so-called random current representation introduced by Aizenman (Comm. Math. Phys., 1982). One of the main contributions of this paper is the development of a corresponding geometric representation for the $φ^4$ model called the random tangled current representation. |
| title | Random tangled currents for $φ^4$: translation invariant Gibbs measures and continuity of the phase transition |
| topic | Probability Mathematical Physics 60K35, 82B20 |
| url | https://arxiv.org/abs/2211.00319 |