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arXiv · 2211.00319

Random tangled currents for $φ^4$: translation invariant Gibbs measures and continuity of the phase transition

Trishen S. Gunaratnam·Christoforos Panagiotis·Romain Panis·Franco Severo
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Abstract

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.

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Trishen S. Gunaratnam, Christoforos Panagiotis, Romain Panis, Franco Severo. 2025-03-23. Random tangled currents for $φ^4$: translation invariant Gibbs measures and continuity of the phase transition. https://arxiv.org/abs/2211.00319

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