Random tangled currents for $\varphi^{4}$: Translation invariant Gibbs measures and continuity of the phase transition

We prove that the set of automorphism invariant Gibbs measures for the \phi^{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 \phi^{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 (2015), and Raoufi (2020) using the so-called random current representation introduced by Aizenman (1982). One of the main contributions of this paper is the development of a corresponding geometric representation for the \phi^{4} model called the random tangled current representation.