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

TL;DR

This paper introduces a new geometric representation called the random tangled current for the $^4$ model, proving results on Gibbs measures and phase transition continuity, extending concepts from the Ising model to $^4$.

Contribution

It develops the random tangled current representation for the $^4$ model and applies it to analyze Gibbs measures and phase transition properties.

Findings

At most two extremal Gibbs measures for the $^4$ model on polynomial growth graphs.

Spontaneous magnetization vanishes at criticality for $^4$ on $Z^d$, $d extgreater 2$.

New geometric representation analogous to the random current for Ising.

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.