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

We prove that any scaling limit of a critical reflection positive Ising or ${\phi }^4$ 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 $1\le d\le 3$ and $d_{\text{eff}}=4$, the phenomenology is different as long-range effects play a prominent role.