Random tangled currents for $\varphi^4$: translation invariant Gibbs measures and continuity of the phase transition
Abstract
We prove that the set of automorphism invariant Gibbs measures for the $\varphi^4$ model on graphs of polynomial growth has at most two extremal measures at all values of $\beta$. 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 $\varphi^4$ 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 $\varphi^4$ model called the random tangled current representation.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2022
- DOI:
- 10.48550/arXiv.2211.00319
- arXiv:
- arXiv:2211.00319
- Bibcode:
- 2022arXiv221100319G
- Keywords:
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- Mathematics - Probability;
- Mathematical Physics;
- 60K35;
- 82B20
- E-Print:
- 92 pages, 4 figures