Triviality of the scaling limits of critical Ising and $\varphi^4$ models with effective dimension at least four

TL;DR

<3-5 sentence high-level summary> The paper proves that at criticality, scaling limits of Ising and $\varphi^4$ models with reflection positivity and effective dimension $d_{\text{eff}}\ge 4$ are Gaussian, extending the earlier near-neighbor results to long-range interactions. It develops a unified framework based on the random current representation, infrared bounds, spectral tools, and a multiscale analysis with a sharp length $L(\beta)$ and regular scales to control intersections of currents. In particular, it establishes Gaussian scaling for $d_{\text{eff}}>4$ and proves Gaussianity in the marginal case $d_{\text{eff}}=4$ via improved diagram bounds and current-geometry arguments; it also extends these results to the Griffiths–Simon class, including the $\varphi^4$ model, through bounds on Ursell functions and the renormalised coupling. The work reinforces universality predictions, clarifies the role of interaction decay in determining scaling limits, and provides tools (regular/mixing bounds, backbone representations) that may aid constructive Euclidean and quantum field theory analyses for high- and marginal-dimensional systems.

Abstract

We prove that any scaling limit of a critical reflection positive Ising or $\varphi^4$ model of effective dimension $d_{\text{eff}}$ at least four is Gaussian. This extends the recent breakthrough work of Aizenman and Duminil-Copin -- which demonstrates the corresponding result in the setup of nearest-neighbour interactions in dimension four -- to the case of long-range reflection positive interactions satisfying $d_{\text{eff}}=4$. The proof relies on the random current representation which provides a geometric interpretation of the deviation of the models' correlation functions from Wick's law. When $d=4$, long-range interactions are handled with the derivation of a criterion that relates the speed of decay of the interaction to two different mechanisms that entail Gaussianity: interactions with a sufficiently slow decay induce a faster decay at the level of the model's two-point function, while sufficiently fast decaying interactions force a simpler geometry on the currents which allows to extend nearest-neighbour arguments. When $1\leq d\leq 3$ and $d_{\text{eff}}=4$, the phenomenology is different as long-range effects play a prominent role.

Figures (7)

• Figure 1: Left: The graph of $\alpha\mapsto d_{\textup{eff}}(\alpha)$ for the interaction $J$ given by $J_{x,y}=C|x-y|_1^{-d-\alpha}$ (for $d\geq 2$). The transition between the regime $d_{\textup{eff}}(\alpha)>d$ and $d_{\textup{eff}}=d$ occurs at $\alpha=2$. At this point, we expect logarithmic corrections at the level of the decay of the critical two-point function. Right: A summary of the expected behaviour of the critical scaling limits for the same interaction $J$. The red region (including segments and points), correspond to interactions which are expected to yield trivial scaling limits. The results of this paper concern the study of the "marginal" cases that separate the two phases.
• Figure 2: An illustration of Proposition \ref{['prop: MMS inequalities general statement']}. The two reflection planes are represented by the green and red dashed lines. The MMS inequalities state that $\langle \tau_0\tau_x\rangle_{\rho,\beta}\geq \langle \tau_0\tau_{\mathcal{P}(x)}\rangle_{\rho,\beta}$ and $\langle \tau_0\tau_y\rangle_{\rho,\beta}\geq \langle \tau_0\tau_{\mathcal{P}^*(y)}\rangle_{\rho,\beta}$.
• Figure 3: A realisation of the event $\mathsf{Jump}(k,m)$ for a current $\mathbf{n}$ with source set $\partial\mathbf{n}=\lbrace 0,y\rbrace$. The bold black path represents the backbone $\Gamma(\mathbf{n})$. The dashed curves represent long open edges that jump over the annulus $\textup{Ann}(k,m)$.
• Figure 4: A graphical representation of the bound \ref{['eq: bound a1 a2 a3']}. We represented each potential contribution to $A_1,A_2,A_3$ (from left to right). The backbone is the bold path joining $0$ and $y$. Long open edges are the dashed curves. The largest contribution should come from $A_2$ in which long edges are induced by the backbone.
• Figure 5: An example of a configuration for which $\mathcal{M}\neq \emptyset$ but $I_k$ does not occur. $\mathbf{n}_1+\mathbf{n}_3$ (resp. $\mathbf{n}_2+\mathbf{n}_4$) is drawn in blue (resp. red), and the backbone of $\mathbf{n}_1$ (resp. $\mathbf{n}_2)$ is the blue (resp. red) bold curve joining $0$ and $y$. The outermost grey region represents the annulus $\mathrm{Ann}(\ell_{k+1},\ell_{k+1}+\ell_{k+1}^\nu)$. The clusters of the origin $\mathbf{C}_{\mathbf{n}_1+\mathbf{n}_3}(0)$ and $\mathbf{C}_{\mathbf{n}_2+\mathbf{n}_4}(0)$ intersect in $\mathrm{Ann}(m,M)$ thanks to a collection of red loops in $(\mathbf{n}_2+\mathbf{n}_4)\setminus \overline{\Gamma(\mathbf{n}_2)}$ which crosses $\mathrm{Ann}(M,N)$, which means that $\mathbf{n}_2+\mathbf{n}_4$ realises $\mathcal{F}_3$. As a result, this intersection is not measurable in terms of edges with both endpoints in $\mathrm{Ann}(\ell_k,\ell_{k+1}+\ell_{k+1}^\nu)$.

Theorems & Definitions (191)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5: Improved tree diagram bound for $d=4$
• Remark 1.6
• Remark 1.7
• Corollary 1.8
• Theorem 1.9: Improved tree diagram bound for $1\leq d \leq 3$
• Remark 1.10