Marginal triviality of the scaling limits of critical 4D Ising and $\phi_4^4$ models

TL;DR

This paper proves that the scaling limits of critical 4D Ising models and $\,\phi_4^4$ fields are Gaussian, showing triviality of their fluctuations through advanced probabilistic and multi-scale analysis techniques.

Contribution

It establishes the Gaussian nature of scaling limits for 4D Ising and $\,\phi_4^4$ models at criticality, using the random current representation and improved bounds.

Findings

Scaling limits are Gaussian at criticality.

Correlation deviations from Wick's law are characterized by intersection probabilities.

Logarithmic correction improves the tree diagram bound.

Abstract

We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the $\lambda \phi^4$ fields over $\mathbb{R}^4$ with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.