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

TL;DR

This paper proves that at the critical point, the scaling limits of certain high-dimensional Ising and $^4$ models are Gaussian, extending previous results to models with long-range interactions satisfying an effective dimension of four.

Contribution

It generalizes the Gaussianity of scaling limits to long-range reflection positive interactions with effective dimension four, beyond nearest-neighbor models.

Findings

Scaling limits are Gaussian for models with $d_{eff} \, \geq \, 4$.

Long-range interactions with specific decay rates influence correlation decay.

Different behaviors are observed in dimensions 1 to 3 versus four.

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.