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

TL;DR

This work provides a comprehensive analysis of the φ^4 model on amenable graphs with polynomial growth, introducing a novel random tangled current representation that extends the random current framework from Ising to unbounded spins. The authors prove a Lebowitz-type structure theorem: for Γ-invariant Gibbs measures at any β, there are at most two extremal measures, and all Γ-invariant Gibbs measures are convex combinations of these plus/minus extremals; a criterion implying uniqueness is given via correlations. They develop a rigorous switching principle for tangled currents, enabling exact correlation inequalities and reducing proofs to percolation-type connectivity properties. A key application shows that spontaneous magnetisation vanishes at criticality under suitable infrared bounds, yielding continuity of the phase transition in high dimensions and for certain decays of interactions; open problems point to 2D and universality questions. The framework combines Griffiths–Simon approximations, regularity and ergodicity arguments, and percolation-style analysis to extend Ising-type results to φ^4 and to general graphs with polynomial growth, with broad implications for universality and critical behavior in non-Gaussian lattice field theories.

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 $β$. 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 $\varphi^4$ 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.

Figures (3)

• Figure 1: On the left: An example of a single tangled current multigraph $\mathcal{H}^A(\mathbf{n},\mathpzc{t})$ with $A_{x_1}=0$, $A_{x_2}=2$, $A_{x_3}=1$ and $A_{x_4}=3$. The circles represent the blocks $\mathcal{B}^A(\mathbf{n})$; the small ellipses inside represent the elements of the partitions $\mathpzc{t}$. The vertices $\{za(k):k\geq 1, \: z\in \Lambda\}$ are represented with squares. Notice that $x_3,x_4$ are sources of $\mathbf{n}$. On the right: An example of a double tangled current multigraph $\mathcal{H}^{A,B}(\mathbf{n}_1,\mathbf{n}_2,\mathpzc{t})$. The vertices in blue are labelled by $(\mathbf{n}_1,A)$ and the ones in red by $(\mathbf{n}_2,B)$. In this example, $A_{x_1}=A_{x_2}=A_{x_3}=0$ and $A_{x_4}=2$; and $B_{x_1}=0$, $B_{x_2}=2$, and $B_{x_3}=B_{x_4}=1$.
• Figure 2: An example illustrating the merging of $\mathcal{C}_{P_i}$ and $\mathcal{C}_{P_j}$.
• Figure 3: An illustration of the mapping used in the proof of Proposition \ref{['prop:bridges']}. On the left, an example of $(\mathbf{n}_1,\mathbf{n}_2,\mathpzc{t})$ satisfying the event $\{\tilde{x} {\centernot\longleftrightarrow} \tilde{y} \text{ in } B_N\}$; on the right, its image $(\mathbf{n}_1',\mathbf{n}_2,\mathpzc{t}')$ by the map. The red vertices represent $\overline{x}, \overline{y}$.

Theorems & Definitions (169)

• Definition 1.1: Gibbs measures
• Theorem 1.2
• Theorem 1.3: Continuity of the phase transition
• Remark 1.4
• Definition 2.1
• Proposition 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Remark 2.5