Marginal triviality of the scaling limits of critical 4D Ising and $\phi _4^4$ models


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 that 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.


Michael Aizenman

Departments of Physics and Mathematics, Princeton University, Princeton, NJ 08544

Hugo Duminil-Copin

Institut des Hautes Études Scientifiques (IHÉS), 91440 Bures-sur-Yvette, France and Université de Genève, 1211 Genève, Switzerland